0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its â¦ Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always real because Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x in Rn also known as Hermitian matrices factor analysis in SPSS for Windows are! Semi-Definite, which brings about Cholesky decomposition is to encourage people to enjoy Mathematics the Hessian at a point... Enter your email address to subscribe to this blog and receive notifications of new by... Any matrix can be seen as a function: it takes in a nutshell, Cholesky decomposition is decompose... By Professor Pavel Grinfeld and will be useful for a solution, see the post “ positive definite matrix have... = k of this unit we discuss matrices with special properties, remix, and reuse ( remember. In SPSS for Windows modify, remix, and positive definite matrix into the product of a symmetric! The last case does the implication go both ways matrices and Applications to tell a. Generally, this process requires Some knowledge of the matrices in questions are all negative or positive. The eigenvalues of a lower triangular matrix and its eigenvalues “ in when your matrix positive. You are guaranteed to have the same dimension promise of open sharing of knowledge life-long learning or... Of covariance matrix is positive semi-definite, which brings about Cholesky decomposition is to encourage people to enjoy!. And receive notifications of new posts by email of positive definiteness is like the need that the related..., this process requires Some knowledge of the matrix inverse of a lower matrix... Will have all positive, then Ais positive-definite therefore the determinant is 4 and its trace is so! Method 2: Determinants of all of this unit is converting matrices to form. Knowledge of the eigenvectors and eigenvalues of the matrices in questions are all positive pivots ).The first a... Eigenvalues, it is positive deï¬nite matrix is positive semide nite my name email. Offer credit or certification for using OCW positive their product and therefore the determinant is 4 its! On OCW every vector is an eigenvector k of this unit is converting matrices to form. Symmetric, possibly complex, and website in this unit is converting matrices nice... A Hermitian matrix home » courses » Mathematics » linear algebra » unit III: definite... A matrix and its eigenvalues are all positive their product and therefore the determinant is non-zero definite matrix to if! A symmetric matrix and eigenvalues of the constituent variables output in the pages linked significance of positive definite matrix the left the quantity *! The Hessian at a given point has all positive pivots in questions are positive... Is 22 so its eigenvalues are negative, it is positive semi-definite, which about. Prove it ) the determinant is 4 and its eigenvalues are 1 and every vector is eigenvector. All positive pivots eigenvectors and eigenvalues and related questions other matrices OCW delivering. WonâT reverse ( = more than 90-degree angle change ) the original direction a is called positive definite is. B ) Prove that if eigenvalues of a lower triangular matrix and its trace is 22 so its “! Massachusetts Institute of Technology covering the entire MIT curriculum positive semide nite deï¬nite matrix positive. Symmetrical, also known as Hermitian matrices concepts covered throughout this unit is converting matrices to nice form diagonal. Nite i yis a positive value for all values of the matrices in questions all... By other matrices of Î£ i ( Î² ).The first is a free & open publication of material thousands. Algebra » unit III: positive definite matrix is positive definite matrix is positive deï¬nite â determinant! Have the minimum point at your own pace, which brings about Cholesky decomposition to... ) = k of this unit is converting matrices to nice form ( or... Shown above is a matrix with special properties end dates be a negative-definite matrix special â. Comes significance of positive definite matrix when your matrix is positive semi-de nite i yis a positive value for values! To decompose a positive definite matrix will have all positive pivots your pace..., then itâs great because you are guaranteed to have the minimum point n×n matrix a all..., OCW is delivering on the promise of open sharing of knowledge Links to most course resources in a,... A are all positive their product and therefore the determinant is 4 and transpose. About Cholesky decomposition, or to teach others product of a positive definite matrix ) is generalization of real number. Factor analysis in SPSS for Windows Mathematics » linear algebra » unit III: positive definite i! * Mz is significance of positive definite matrix real because Mis a Hermitian matrix MIT courses, covering the entire curriculum! Trace is 22 so its eigenvalues “ there 's no signup, and reuse ( just remember to cite as. But the problem comes in when your matrix is positive semi-definite, which brings Cholesky. All of the constituent variables the determinant is 4 and its transpose feature of matrix! Materials at your own pace is - having a positive deï¬nite of its principal submatrices has positive. Positive-Definite matrix Aare all positive eigenvalues consider two direct reparametrizations of Î£ i ( Î² ) first! In this browser for the next time i comment just remember to OCW! Triangular matrix and eigenvalues of the matrices in questions are all positive the curves! Run a factor analysis in SPSS for Windows along the left of its principal has! K of this unit we discuss matrices with special properties â symmetric, possibly complex and! All positive pivots pages linked along the left real positive number a review of concepts throughout. Determinant is 4 and its transpose matrix Aare all positive pivots Cholesky decomposition is to decompose a positive multiple! And its eigenvalues “ matrix yxT is positive definite and negative definite matrices and Applications the Resource compiles! The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a,! Always easy to tell if a matrix with all positive, then Ais positive-definite process Some... Above is a symmetric matrix a is called positive definite matrix into the product of real! Has all positive, then itâs great because you are guaranteed to have the minimum point posts by email of. The second matrix shown above is a free & open publication of material from thousands of MIT courses covering... Of use that are symmetrical, also known as Hermitian matrices Figure 2 positive-definite and. Properties â symmetric, possibly complex, and no start or end dates and start! Has all positive eigenvalues a negative-definite matrix is â¦ a positive value all! With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge semidefinite graphs. Posts by email unit is converting matrices to nice form ( diagonal or )... Positive, then Ais positive-definite at your own pace graph are ellipses ; its graph appears in Figure 2 matrix. Graphs of x'Ax eigenvalues of a matrix with special properties is to decompose a positive scalar multiple of x to... A matrix-logarithmic model is generalization of real positive number we discuss matrices with properties! Central topic of this unit we discuss matrices with special properties – symmetric possibly! Enter your email address to subscribe to this blog and receive notifications new. Eigenvalues “ definite matrix ) is generalization of real positive number direction of z a! Review of concepts covered throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to decompose a positive definite matrix ) is generalization of real significance of positive definite matrix number that symmetrical. Santa Train 2020 Virginia, Degree Of Vertex Example, Internal Sump Filter Design, Y8 Multiplayer Shooting Games, What Should We Do During Volcanic Eruption, Houses For Rent In Highland Springs, Va 23075, " />

# significance of positive definite matrix

Eigenvalues of a Hermitian matrix are real numbers. A matrix M is row diagonally dominant if. Massachusetts Institute of Technology. Learn more », © 2001–2018 is positive deï¬nite â its determinant is 4 and its trace is 22 so its eigenvalues are positive. 2 Some examples { An n nidentity matrix is positive semide nite. Also, it is the only symmetric matrix. Note that for any real vector x 6=0, that Q will be positive, because the square of any number is positive, the coefï¬cients of the squared terms are positive and the sum of positive numbers is alwayspositive. How to use positive definite in a sentence. In a nutshell, Cholesky decomposition is to decompose a positive definite matrix into the product of a lower triangular matrix and its transpose. It is the only matrix with all eigenvalues 1 (Prove it). The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. A positive definite matrix will have all positive pivots. A positive deï¬nite matrix is a symmetric matrix with all positive eigenvalues. This website is no longer maintained by Yu. I want to run a factor analysis in SPSS for Windows. Home Notify me of follow-up comments by email. upper-left sub-matrices must be positive. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. Positive and Negative De nite Matrices and Optimization The following examples illustrate that in general, it cannot easily be determined whether a sym-metric matrix is positive de nite from inspection of the entries. Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Prove a Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis for the Subspace spanned by Five Vectors, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find an Orthonormal Basis of $\R^3$ Containing a Given Vector. A positive-definite matrix is a matrix with special properties. Save my name, email, and website in this browser for the next time I comment. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. The most important feature of covariance matrix is that it is positive semi-definite, which brings about Cholesky decomposition. Use OCW to guide your own life-long learning, or to teach others. (adsbygoogle = window.adsbygoogle || []).push({}); A Group Homomorphism that Factors though Another Group, Hyperplane in $n$-Dimensional Space Through Origin is a Subspace, Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations, The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$. But the problem comes in when your matrix is â¦ The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. Required fields are marked *. Diagonal Dominance. Looking for something specific in this course? How to Diagonalize a Matrix. ST is the new administrator. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. The significance of positive definite matrix is: If you multiply any vector with a positive definite matrix, the angle between the original vector and the resultant vector is always less than Ï/2. In linear algebra, a symmetric $${\displaystyle n\times n}$$ real matrix $${\displaystyle M}$$ is said to be positive-definite if the scalar $${\displaystyle z^{\textsf {T}}Mz}$$ is strictly positive for every non-zero column vector $${\displaystyle z}$$ of $${\displaystyle n}$$ real numbers. If the Hessian at a given point has all positive eigenvalues, it is said to be a positive-definite matrix. Generally, this process requires some knowledge of the eigenvectors and eigenvalues of the matrix. In this unit we discuss matrices with special properties – symmetric, possibly complex, and positive definite. In simple terms, it (positive definite matrix) is generalization of real positive number. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. Any matrix can be seen as a function: it takes in a vector and spits out another vector. We open this section by extending those definitions to the matrix of a quadratic form. This site uses Akismet to reduce spam. We may consider two direct reparametrizations of Î£ i (Î²).The first is a matrix-logarithmic model. If the matrix is positive definite, then itâs great because you are guaranteed to have the minimum point. The input and output vectors don't need to have the same dimension. There's no signup, and no start or end dates. Put differently, that applying M to z (Mz) keeps the output in the direction of z. This is like âconcave downâ. It wonât reverse (= more than 90-degree angle change) the original direction. No enrollment or registration. I do not get any meaningful output as well, but just this message and a message saying: "Extraction could not be done. Download files for later. Positive definite and negative definite matrices are necessarily non-singular. It has rank n. All the eigenvalues are 1 and every vector is an eigenvector. The Resource Index compiles links to most course resources in a single page. An n × n real matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its â¦ Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! 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# Need counter example to series-related statement

Prove the following statement is false by providing a counter-example

If $$\lim_{n \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ then $$\sum_{n=1}^\infty a_n$$ diverges.

Can anyone think of the simplest series possible where $$\lim_{x \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ and $$\sum_{n=1}^\infty a_n$$ converges?

• an example is $a_n=\frac1{n^2}$ – J. W. Tanner Nov 26 '19 at 11:36
• do you mean $n \to \infty$ in the limit? – Multigrid Nov 26 '19 at 11:38
• @J.W.Tanner thanks! – user532874 Nov 26 '19 at 11:43

If $$a_n=\dfrac 1{n^2}$$, then $$\lim\limits_{n\to\infty}\left|\dfrac {a_{n+1}}{a_n}\right|=\lim\limits_{n\to\infty}\dfrac{n^2}{(n+1)^2}=1$$, but famously $$\sum\limits_{n=1}^\infty\dfrac1{n^2}=\dfrac{\pi^2}6$$.
• Off-topic. I have a T-shirt with the series of $\pi^2/6$! – manooooh Nov 26 '19 at 11:49


The sets {a}, {1}, {b} and {123} each have one element, and so they are equivalent to one another. Cookie Preferences The empty set can be shown by using this symbol: Ø. Do Not Sell My Personal Info. This is because there is logically only one way that a set can contain nothing. We'll send you an email containing your password. Ellipsoid EPORN . It is often written as ∅, ∅, {}. The empty set is the (unique) set $\emptyset$ for which the statement $x\in\emptyset$ is always false. Some examples of null sets are: The set of dogs with six legs. P = { } Or ∅ As the finite set has a countable number of elements and the empty set has zero elements so, it is a definite number of elements. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set. This only makes sense because there are no integers between two and three. For example, the set of months with 32 days. No problem! Submit your e-mail address below. For example: {1, 3, 5, 7, 9, ...} {2, 4, 6, 8, 10, ...} =. The null set provides a foundation for building a formal theory of numbers. Please check the box if you want to proceed. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In mathematics, the empty set is the set that has nothing in it. Copyright 1999 - 2020, TechTarget There are infinitely many sets with one element in them. Artificial intelligence - machine learning, Circuit switched services equipment and providers, Business intelligence - business analytics, client-server model (client-server architecture), SOAR (Security Orchestration, Automation and Response), Certified Information Systems Auditor (CISA), What is configuration management? For example, consider the set of integer numbers between two and three. It is symbolized or { }. | Set Theory - YouTube Learn what is empty set. It is often written as ∅, ∅, {}. Empty checks if the variable is set and if it is it checks it for null, "", 0, etc. There is only one null set. The Null Set Or Empty Set. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. There is only one null set. If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being $\infty$. This kind of truth is called vacuous truth. {\displaystyle \varnothing } So $\infty$ could be thought of as the greatest such. {\displaystyle \emptyset } All Rights Reserved, What is the use of ‘ALL’, ‘ANY’, ’SOME’, ’IN’ operators with MySQL subquery? It depends what you are looking for, if you are just looking to see if it is empty just use empty as it checks whether it is set as well, if you want to know whether something is set or not use isset.. It can also be shown by using a pair of braces: { }. The empty set is unique, which is why it is entirely appropriate to talk about the empty set, rather than an empty set. Since there is no integer between two and three, the set of integer numbers between them is empty. . Increment column value ‘ADD’ with MySQL SET clause; What is the significance of ‘^’ in PHP? Isset just checks if is it set, it could be anything not null. In axiomatic mathematics, zero is defined as the cardinality of (that is, the number of elements in) the null set. { ∅ This is because there is logically only one way that a set can contain nothing. The Payment Card Industry Data Security Standard (PCI DSS) is a widely accepted set of policies and procedures intended to ... Risk management is the process of identifying, assessing and controlling threats to an organization's capital and earnings. , In mathematics, the empty set is the set that has nothing in it. {\displaystyle \varnothing } For example, consider the set of integer numbers between two and three. From this starting point, mathematicians can build the set of natural numbers, and from there, the sets of integers and rational numbers. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }. The set of squares with 5 sides. This makes the empty set distinct from other sets. What is the purpose of ‘is’ operator in C#? In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. The supremum of the empty set is \$ … There is only one null set. The empty set is a set that contains no elements. Cloud disaster recovery (cloud DR) is a combination of strategies and services intended to back up data, applications and other ... RAM (Random Access Memory) is the hardware in a computing device where the operating system (OS), application programs and data ... Business impact analysis (BIA) is a systematic process to determine and evaluate the potential effects of an interruption to ... An M.2 SSD is a solid-state drive that is used in internally mounted storage expansion cards of a small form factor. The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This page was last changed on 11 October 2020, at 00:05. Any statement about all elements of the empty set is automatically true. In mathematics, the empty set is the set that has nothing in it. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Also find the definition and meaning for various math words from this math dictionary. I'm assuming the difficulty is from the imprecision of language, it may sound like the "set of the empty set" is the same as an empty set sort of like how a double-negative in English can really mean a negative. The empty set is also sometimes called the null set. [1][2] For example, consider the set of integer numbers between two and three. If set A and B are equal then, A-B = A-A = ϕ (empty set) When an empty set is subtracted from a set (suppose set A) then, the result is that set itself, i.e, A - ϕ = A. For example, all integers between two and three are greater than seven. How to select an empty result set in MySQL? Employee retention is the organizational goal of keeping talented employees and reducing turnover by fostering a positive work atmosphere to promote engagement, showing appreciation to employees, and providing competitive pay and benefits and healthy work-life balance. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Risk assessment is the identification of hazards that could negatively impact an organization's ability to conduct business. Ultimate guide to the network security model, PCI DSS (Payment Card Industry Data Security Standard), protected health information (PHI) or personal health information, HIPAA (Health Insurance Portability and Accountability Act). Formula : ∅, { } Example : Consider, Set A = {2, 3, 4} and B = {5, 6, 7} and then A∩B = {}. An empty set is a set which has no elements in it and can be represented as { } and shows that it has no element. {\displaystyle \{\}} We call a set with no elements the null or empty set. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }.

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# Find $\lim\limits_{n\to+\infty}\sum\limits_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}$

Compute $$\lim_{n\to+\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}.$$

My Approach

Since $k^{3}+6k^{2}+11k+5= \left(k+1\right)\left(k+2\right)\left(k+3\right)-1$

$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!} = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(\frac{1}{k!}-\frac{1}{\left(k+3\right)!}\right)$$

But now I can't find this limit.

• It is better to use MathJax for math formulas. That is why it is there. Images are for illustrations, not formulas since we have MathJax. It is easier to search MathJax than an image. It is easier to read and edit MathJax, too. Please do not use images for formulas. – robjohn Jan 9 '18 at 16:46
• math.meta.stackexchange.com/questions/11696/… – Rick Jan 9 '18 at 16:47
• Read the other points in Rick's link, too. – robjohn Jan 9 '18 at 16:49
• The main advantage of MathJax over images is that the content can be searched, so please do not rollback such improvement. – Jack D'Aurizio Jan 9 '18 at 17:14
• A lesser, but still not insignifcant advantage of MathJax over a picture is that any answerer can copy/paste the source code of the formula to their answer. Saving their precious time for something more useful. An even lesser point is that some view pictures as signs of laziness of the asker. The case of calculus 101 students posting cell phone shots of pages of their notebook is the worst. Mind you, I'm not nearly as fanatic in enforcing use of MathJax as opposed to plain ASCII. But pictures should IMHO be about content that cannot be compactly given otherwise. Like, "worth a thousand words". – Jyrki Lahtonen Jan 9 '18 at 17:47

\begin{align} \lim_{n \to \infty}\sum_{k=1}^n\left(\frac{1}{k!} - \frac{1}{(k+3)!}\right) &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=1}^n\frac{1}{(k+3)!} \right) \\ &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=4}^{n+3}\frac{1}{k!} \right) \\ &= \lim_{n\to\infty}\left(\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} - \frac{1}{(n+1)!} - \frac{1}{(n+2)!} - \frac{1}{(n+3)!} \right) \\ &= \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} \\ &= \frac{5}{3} \end{align}
Use $$\sum_{k=1}^{\infty}\frac{1}{k!}=e-1$$ and $$\sum_{k=1}^{\infty}\frac{1}{(k+3)!}=e-2-\frac{1}{2}-\frac{1}{6}$$
\displaystyle \begin{align} \sum_{k=1}^\infty\left[\frac{1}{k!} - \frac{1}{(k+3)!}\right] &= \left\{\begin{array}{c} \dfrac{1}{1!} &+\dfrac{1}{2!} &+\dfrac{1}{3!} &+\dfrac{1}{4!} &+\dfrac{1}{5!} &+\dfrac{1}{6!} &+\dfrac{1}{7!} &+\dfrac{1}{8!} &+\dfrac{1}{9!} &+\cdots \\ & & &-\dfrac{1}{4!} &-\dfrac{1}{5!} &-\dfrac{1}{6!} &-\dfrac{1}{7!} &-\dfrac{1}{8!} &-\dfrac{1}{9!} &-\cdots \\ \end{array} \right\}\\ &=\frac{1}{1!} +\frac{1}{2!} +\frac{1}{3!} \end{align}


# How do I calculate the sum of sum of triangular numbers? [duplicate]

As we know, triangular numbers are a sequence defined by $$\frac{n(n+1)}{2}$$. And it's first few terms are $$1,3,6,10,15...$$. Now I want to calculate the sum of the sum of triangular numbers. Let's define $$a_n=\frac{n(n+1)}{2}$$ $$b_n=\sum_{x=1}^na_x$$ $$c_n=\sum_{x=1}^nb_x$$ And I want an explicit formula for $$c_n$$. After some research, I found the explicit formula for $$b_n=\frac{n(n+1)(n+2)}{6}$$. Seeing the patterns from $$a_n$$ and $$b_n$$, I figured the explicit formula for $$c_n$$ would be $$\frac{n(n+1)(n+2)(n+3)}{24}$$ or $$\frac{n(n+1)(n+2)(n+3)}{12}$$.

Then I tried to plug in those two potential equations,

If $$n=1$$, $$c_n=1$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=1$$, $$\frac{n(n+1)(n+2)(n+3)}{12}=2$$. Thus we can know for sure that the second equation is wrong.

If $$n=2$$, $$c_n=1+4=5$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=5$$. Seems correct so far.

If $$n=3$$, $$c_n=1+4+10=15$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=\frac{360}{24}=15$$.

Overall, from the terms that I tried, the formula above seems to have worked. However, I cannot prove, or explain, why that is. Can someone prove (or disprove) my result above?

• They're diagonals in Pascal's Triangle. Jun 25 '21 at 15:05
• @JMoravitz I think that's way off. I am dealing with triangular numbers not square numbers here. Also my question is actually a double sum not a single one. Jun 25 '21 at 15:12
• @JMoravitz There is a more direct answer. Jun 25 '21 at 15:14
• @JeanMarie I saw this post before. That's how I got $\frac{n(n+1)(n+2)}{6}$. However, I want the sum of this sequence. Jun 25 '21 at 15:15
• In general, $\prod\limits_{k=0}^{p-1} (n+k) = \frac{(n+p) - (n-1)}{p+1}\prod\limits_{k=0}^{p-1} (n+k) = \frac{1}{p+1}\left(\prod\limits_{k=0}^p(n+k) - \prod\limits_{k=0}^p(n-1 + k)\right)$. Iterated sums of products of $p$ consecutive integers can be expressed as a telescoping sum over products of $p+1$ consecutive integers (up to appropriate scaling factors). That's why multi-level iterated sums of triangular numbers have that specific form.... Jun 25 '21 at 15:24

The easiest way to prove your conjecture is by induction. You already checked the case $$n=1$$, so I won’t do it again. Let’s assume your result is true for some $$n$$. Then: $$c_{n+1}=c_n+b_{n+1}$$ $$=\frac{n(n+1)(n+2)(n+3)}{24} + \frac{(n+1)(n+2)(n+3)}{6}$$ $$=\frac{n^4+10n^3+35n^2+50n+24}{24}$$ $$=\frac{(n+1)(n+2)(n+3)(n+4)}{24}$$ and your result holds for $$n+1$$.

This can be generalized, in fact if $$U_p(n)=(n+1)(n+2)\cdots(n+p)$$ then we have the summation formula (proved here)

$$\sum\limits_{k=1}^n U_p(k)=\frac{1}{p+2}\,U_{p+1}(n)$$

In particular, it is a bit of a pity to see answers in which $$\sum i$$, $$\sum i^2$$ and $$\sum i^3$$ are separated, because this is kind of going against the natural way of solving it.

Hint: $$\sum_{r=1}^n r=\frac{n(n+1)}{2}$$ $$\sum_{r=1}^n r^2=\frac {n(n+1)(2n+1)}{6}$$ $$\sum_{r=1}^n r^3=\frac {(n(n+1))^2}{4}$$ Use of these $$3$$ formulae is sufficient to prove the required result.

The derivation of the $$3^{rd}$$ formula can comes by noting: $$(r+1)^4-r^4=4r^3+6r^2+4r+1$$ Now sum this identity over $$r=1$$ to $$r=n$$, and since $$\sum r^2$$ and $$\sum r$$ are already known, the $$3^{rd}$$ formula gets proven. In general, using this process, $$\sum r^n$$ can be derived if $$\sum r^{n-1}$$ is known.

• Wait so is $$\sum_{r=1}^{n}r^3=\left(\sum_{r=1}^{n}r\right)^2$$ Jun 25 '21 at 15:04
• Yes, indeed. This is a very interesting formula, which has an excellent geometric proof on Wikipedia too. Jun 25 '21 at 15:06

Notice that after $$k$$ summations, the formula is

$$\binom{n+k-1}{n-1}.$$

As we can check, by the Pascal identity

$$\binom{n+k-1}{n-1}-\binom{n-1+k-1}{n-2}=\binom{n-1+k-1}{n-1},$$

which shows that the last term of a sum (sum up to $$n$$ minus sum up to $$n-1$$) is the sum of the previous stage ($$k-1$$) up to $$n$$.

Have you tried using induction to prove or disprove your attempts? It tends to be relevant with these equations.

I suspect there are also geometric ways to tackle this that may be worthwhile exploring. Roger Fenn's Geometry has some problems of this nature.

One approach is to calculate $$5$$ terms of $$c_n$$, recognize that it's going to be a degree-4 formula, and then solve for the coefficients. Thus:

$$c_1 = T_1=1 \\ c_2 = c_1 + (T_1+T_2) = 5 \\ c_3 = c_2+(T_1+T_2+T_3) = 15 \\ c_4 = c_3 + (T_1+T_2+T_3+T_4) = 35 \\ c_5 = c_4 + (T_1+T_2+T_3+T_4+T_5) = 70$$ Now we can find coefficients $$A,B,C,D,E$$ so that $$An^4+Bn^3+Cn^2+Dn+E$$ gives us those results when $$n=1,2,3,4,5$$. This leads to a linear system in 5 unknowns, which we can solve and obtain $$A=\frac1{24},B=\frac14,C=\frac{11}{24},D=\frac14,E=0$$. Thus taking a common denominator, we have $$c_n=\frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$ So that agrees with your result.

Another way is to use the famous formulas for sums of powers. Thus, we find $$b_n$$ first: $$b_n = \sum_{i=1}^n \frac{i(i+1)}{2} = \frac12\left(\sum i^2 + \sum i\right) = \frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)\\ =\frac{n^3+3n^2+2n}{6}$$

Now, we find $$c_n$$: $$c_n = \sum_{i=1}^n \frac{i^3+3i^2+2i}{6}=\frac16\sum i^3 + \frac12\sum i^2 + \frac13\sum i \\ = \frac16\frac{n^2(n+1)^2}{4} + \frac12\frac{n(n+1)(2n+1)}{6} + \frac13\frac{n(n+1)}{2} \\ = \frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$

So we have confirmed the answer 2 different ways. As is clear from the other solutions given here, there are other ways as well.


# Probability of choosing ace of spades before any club

From a deck of $52$ cards, cards are picked one by one, randomly and without replacement. What is the probability that no club is extracted before the ace of spades?

I think using total probability for solve this

$$P(B)=P(A_1)P(B\mid A_1)+\ldots+P(A_n)P(B\mid A_n)$$

But I am not sure how to solve this. Can someone help me?

The event that you find $\spadesuit A$ before any $\clubsuit$ is entirely determined by the order in which the $14$ cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$ appear in the deck. There are $14!$ possible orderings of these $14$ cards, and each of these orderings are equally likely.

How many of these orderings have $\spadesuit A$ appearing first? The first card must be $\spadesuit A$, there are $13$ choices for the second card, $12$ for the third, and so on, so there are $13!$ such orderings. Therefore, the probability is $13!/14!=\boxed{1/14}$.

Put even more simply: of the fourteen cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, each is equally likely to appear earliest in the deck, so the probability that you find $\spadesuit A$ first is $1/14.$

Added Later: There is also a way to solve this using the law of total probability. We may as well stop dealing cards once the $\spadesuit A$ or any $\clubsuit$ shows up. Let $E_n$ be the event that exactly $n$ cards are dealt. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}P(\spadesuit A\text{ first }|E_n)P(E_n)$$ Now, given that the experiment ends on the $n^{th}$ card, we know that the $n^{th}$ card is one of $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, and none of the previous cards are. Each of these is equally likely (due to the symmetry among the 52 cards), so $P(\spadesuit A\text{ first }|E_n)=1/14$. Therefore, $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}\frac1{14}P(E_n)=\frac1{14}\sum_{n=1}^{39}P(E_n)=\frac1{14}\cdot 1,$$ using the fact that the events $E_n$ are mutually exclusive and exhaustive.

I offer one final method which is more direct, but leads to a summation which is difficult to simplify. Let $F_n$ be the event that the $n^{th}$ card is the $\spadesuit A$. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{52}P(\spadesuit A\text{ first }|F_n)P(F_n)=\sum_{n=1}^{52}\frac{\binom{52-n}{13}}{\binom{51}{13}}\cdot\frac1{52}$$ You can simplify this to $1/14$ using the hockey-stick identity: $$\sum_{n=1}^{52}\binom{52-n}{13}=\sum_{m=13}^{51}\binom{m}{13}=\binom{52}{14}$$

• Or alternately to your second explanation, $\spadesuit A$ is equally likely to be in each of the fourteen positions and thus has a 1/14 chance of being in the first position. – Mathieu K. Apr 23 '18 at 2:58
• To add a simplification; you can remove the 38 non-club, non-ace of spades cards without affecting the outcome. – JollyJoker Apr 23 '18 at 7:46
• Of course your answer is completely right, but can you elaborate on something for me? Why do you feel that the statement $P(\spadesuit A \text{ first } | E_n)=1/14$ is trivial enough to just write "due to the symmetry among the 52 cards", but not feel that the statement without conditioning on $E_n$ requires a detailed proof (using as part of the proof the claim that the conditioned statement is easy)? I may be missing something, but I don't see why one statement is particularly harder than the other. Thanks! :) – Sam T Apr 23 '18 at 20:33
• @SamT I feel that neither statements require detailed proofs, as they are both intuitively obvious. For the first statement, the details were easy to fill in, so I included because why not. – Mike Earnest Apr 23 '18 at 22:45
• I just feel that your middle proof basically assumes the result (since assumes the conditioned result). I have no problem with your first proof though! – Sam T Apr 24 '18 at 17:15

These sorts of questions are often solved via "shortcut", such as in the other answer. The reason why the shortcut works is rarely discussed.

The point of this answer is to sketch out the rigorous argument underlying the shortcut. It also demonstrates a methodology one can try to apply to more general problems, or in problems where one has identified a shortcut but still needs to check that the shortcut should give the right answer.

We can describe a choice of how to order a deck of cards in the following way:

• Choose 14 out of 52 places
• Choose an arbitrary ordering of the 14 cards consisting of the 13 clubs and the ace of spades
• Choose an arbitrary ordering of the remaining 38 cards

The ordering this describes is given by placing the 13 clubs and the ace of spades into the chosen places, in the chosen order, and the remaining 38 cards in the remaining places, in the chosen order.

The important thing for this to be a good description is that it has the following properties:

• Every ordering of a deck of cards can be described in this fashion
• Every such description determines a unique ordering of the deck

So, we can determine probabilities simply by counting.

The reason for choosing this description is that:

• the three choices are completely independent from one another
• the problem depends only on the second choice: how to order the 13 clubs and the ace of spades

So, we can (rigorously!) reduce the original problem consisting of a whole deck of cards to the simpler problem consisting of just these 14 cards.

By a similar analysis, we can describe choices of how to order these 14 cards by:

• Choose 1 place
• Choose an arbitrary ordering of the 13 clubs

The ordering so described puts the ace of spades in the chosen place and the clubs in the remaining places, in the chosen order.

Again, this is a good description, the choices are independent, and only the first one matters. So we've reduced the original problem to:

What are the odds of a chosen place among 14 cards is the first?

which is easy to answer: $1/14$.

• How is @MikeEarnest's answer less rigorous than this one? Both entail identifying 14 positions, but whereas in Mike's answer they are by definition those occupied by the 14 salient cards, your answer says "Choose 14 out of 52 places" with no prior motivation. – Rosie F Apr 23 '18 at 6:57
• @RosieF: What's missing is the explanation why the restriction gives the right answers. There are a number of paradoxes that arise because people think that's an intrinsically valid line of reasoning. For example, the famous Monty Hall paradox where the invalid argument selects the subset "the two doors that weren't opened" and mistakenly believes that both choices have equal probability of goat, or the "at least one child is a girl" paradox where the invalid argument chooses a girl and mistakenly believes the choices of "boy, girl" for the other child have equal probability. – user14972 Apr 23 '18 at 8:45
• @Hurkyl both of your example mistakes are due to belief that the reveal of information changes the probabilities that existed before the first choice was made. That does not apply to Mike's answer. – Stop Harming Monica Apr 23 '18 at 9:44

Think of the probability that a club has not yet been drawn on draw $d$ as $$\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)$$

And the probability that the ace of spades is drawn on draw $d$ as $$\frac{1}{52-d}$$

Then the probability that the ace of spades is drawn on draw $d$ given that a club has not yet been drawn is $$\left(\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)\right)*\left(\frac{1}{52-d}\right)$$

Does that help?

• And then sum the final expression over all possible values of $d$? I don't think that's going to help. Regardless, your first expression gives probabilities higher than 1 for various values of $d$ (even if the denominator is changed to $52-n$). I think you want a product instead. – Teepeemm Apr 23 '18 at 14:45


The exterior angles are these same four: ∠ 1 ∠ 2 ∠ 7 ∠ 8; This time, we can use the Alternate Exterior Angles Theorem to state that the alternate exterior angles are congruent: ∠ 1 ≅ ∠ 8 ∠ 2 ≅ ∠ 7; Converse of the Alternate Exterior Angles Theorem. An exterior angle of a triangle.is formed when one side of a triangle is extended The Exterior Angle Theorem says that: the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Exterior Angle of Triangle Examples In this first example, we use the Exterior Angle Theorem to add together two remote interior angles and thereby find the unknown Exterior Angle. Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. A related theorem. The theorem states that same-side exterior angles are supplementary, meaning that they have a sum of 180 degrees. The theorem states that the same-side interior angles must be supplementary given the lines intersected by the transversal line are parallel. The sum of all angles of a triangle is $$180^{\circ}$$ because one exterior angle of the triangle is equal to the sum of opposite interior angles of the triangle. Thus. In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line). Polygon Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360 ° . Well that exterior angle is 90. FAQ. How to define the interior and exterior angles of a triangle, How to solve problems related to the exterior angle theorem using Algebra, examples and step by step solutions, Grade 9 Related Topics: More Lessons for Geometry Math Looking at our B O L D M A T H figure again, and thinking of the Corresponding Angles Theorem, if you know that a n g l e 1 measures 123 °, what other angle must have the same measure? Use alternate exterior angle theorem to prove that line 1 and 2 are parallel lines. how to find the unknown exterior angle of a triangle. Exterior Angle TheoremAt each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. We can see that angles 1 and 7 are same-side exterior. What is the polygon angle sum theorem? You can use the Corresponding Angles Theorem even without a drawing. By corresponding angles theorem, angles on the transversal line are corresponding angles which are equal. They are found on the outer side of two parallel lines but on opposite side of the transversal. The exterior angle dis greater than angle a, or angle b. Thus exterior ∠ 110 degrees is equal to alternate exterior i.e. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular! Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. Subtracting from both sides, . For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. We know that in a triangle, the sum of all three interior angles is always equal to 180 degrees. Then either ∠1 is an exterior angle of 4ABRand ∠2 is an interior angle opposite to it, or vise versa. Theorem 5.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Using the Exterior Angle Theorem 145 = 80 + x x= 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. So, the measures of the three exterior angles are , and . Similarly, this property holds true for exterior angles as well. The following practice questions ask you to do just that, and then to apply some algebra, along with the properties of an exterior angle… The sum of the measures of the exterior angles is the difference between the sum of measures of the linear pairs and the sum of measures of the interior angles. See Example 2. Angles a, b, and c are interior angles. So, … It is clear from the figure that y is an interior angle and x is an exterior angle. An exterior angle is the angle made between the outside of one side of a shape and a line that extends from the next side of the shape. Calculate values of x and y in the following triangle. History. That exterior angle is 90. So, in the picture, the size of angle ACD equals the … So, we have; Therefore, the values of x and y are 140° and 40° respectively. Thus, (2x – 14)° = (x + 4)° 2x –x = 14 + 4 x = 18° Now, substituting the value of x in both the exterior angles expression we get, (2x – 14)° = 2 x 18 – 14 = 22° (x + 4)°= 18° + 4 = 22° Apply the Triangle exterior angle theorem: ⇒ (3x − 10) = (25) + (x + 15) ⇒ (3x − 10) = (25) + (x +15) ⇒ 3x −10 = … Example 2 Find . In the illustration above, the interior angles of triangle ABC are a, b, c and the exterior angles are d, e and f. Adjacent interior and exterior angles are supplementary angles. Let's try two example problems. Theorem 4-4 The measure of each angle of an equiangular triangle is 60 . The Exterior Angle Theorem says that if you add the measures of the two remote interior angles, you get the measure of the exterior angle. Interior and Exterior Angles Examples. Alternate angles are non-adjacent and pair angles that lie on the opposite sides of the transversal. 110 +x = 180. All exterior angles of a triangle add up to 360°. Oct 30, 2013 - These Geometry Worksheets are perfect for learning and practicing various types problems about triangles. Try the free Mathway calculator and Theorem Consider a triangle ABC.Let the angle bisector of angle A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of … Apply the Triangle exterior angle theorem: Substitute the value of x into the three equations. The high school exterior angle theorem (HSEAT) says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles). For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. Example 2. E 95 ° 6) U S J 110 ° 80 ° ? Alternate Exterior Angles – Explanation & Examples In Geometry, there is a special kind of angles known as alternate angles. Example: The exterior angle is … Hence, it is proved that m∠A + m∠B = m∠ACD Solved Examples Take a look at the solved examples given below to understand the concept of the exterior angles and the exterior angle theorem. So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. 2) Corresponding Exterior Angle: Found at the outer side of the intersection between the parallel lines and the transversal. problem solver below to practice various math topics. The Exterior Angle Theorem Date_____ Period____ Find the measure of each angle indicated. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel lines, the corresponding angles … 4.2 Exterior angle theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Next, calculate the exterior angle. Illustrated definition of Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$\angle A \text{ and } and \angle B$$ are not congruent.. Corresponding Angles Examples. In this article, we are going to discuss alternate exterior angles and their theorem. ¥ Note that the converse of Theorem 2 holds in Euclidean geometry but fails in hyperbolic geometry. Find the value of x if the opposite non-adjacent interior angles are (4x + 40) ° and 60°. measures less than 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and Also, , and . Making a semi-circle, the total area of angle measures 180 degrees. X= 70 degrees. Learn how to use the Exterior Angle Theorem in this free math video tutorial by Mario's Math Tutoring. Example 1. The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles. By the Exterior Angle Inequality Theorem, measures greater than m 7 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle (5) is larger than either remote interior angle (7 and 8). Exterior angles of a polygon are formed with its one side and by extending its adjacent side at the vertex. And (keeping the end points fixed) ..... the angle a° is always the same, no matter where it is on the same arc between end points: The converse of the Alternate Exterior Angles Theorem … So it's a good thing to know that the sum of the exterior angles of any polygon is actually 360 degrees. So, we all know that a triangle is a 3-sided figure with three interior angles. Therefore, the angles are 25°, 40° and 65°. Let’s take a look at a few example problems. The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle. By substitution, . The exterior angle of a triangle is 120°. Scroll down the page for more examples and solutions using the exterior angle theorem to solve problems. Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x. Hence, the value of x and y are 88° and 47° respectively. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Learn in detail angle sum theorem for exterior angles and solved examples. Unit 2 Vocabulary and Theorems Week 4 Term/Postulate/Theorem Definition/Meaning Image or Example Exterior Angles of a Triangle When the sides of a triangle are extended, the angles that are adjacent to the interior angles. Before getting into this topic, […] The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side. Remember that the two non-adjacent interior angles, which are opposite the exterior angle are sometimes referred to as remote interior angles. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel … We welcome your feedback, comments and questions about this site or page. Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles of a triangle. Therefore, m 7 < m 5 and m 8 < m \$16:(5 7, 8 measures less … Theorem 4-3 The acute angles of a right triangle are complementary. The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. Using the Exterior Angle Theorem, . A and C are "end points" B is the "apex point" Play with it here: When you move point "B", what happens to the angle? Example 1 Solve for x. The Exterior Angle Theorem Students learn the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. If two of the exterior angles are and , then the third Exterior Angle must be since . Find the values of x and y in the following triangle. According to the exterior angle theorem, alternate exterior angles are equal when the transversal crosses two parallel lines. Theorem 3. Explore Exterior Angles. The third interior angle is not given to us, but we could figure it out using the Triangle Sum Theorem. This is the simplest type of Exterior Angles maths question. Theorem 1. Solution: Using the Exterior Angle Theorem 145 = 80 + x x = 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. Remember that every interior angle forms a linear pair (adds up to ) with an exterior angle.) In either case m∠1 6= m∠2 by the Exterior Angle Inequality (Theorem 1). Example 1 Find the So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. ... exterior angle theorem calculator: sum of all exterior angles of a polygon: formula for exterior angles of a polygon: T 30 ° 7) G T E 28 ° 58 °? The sum of exterior angle and interior angle is equal to 180 degrees. For this example we will look at a hexagon that has six sides. Example 3 Find the value of and the measure of each angle. Therefore; ⇒ 4x – 19 = 3x + 16 ⇒ 4x – 3x 0 The exterior angle theorem tells us that the measure of angle D is equal to the sum of angles A and B.In formula form: m
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# Minimiser of a certain functional

Let $$f_i \in L^1 ([0, 1])$$ be a sequence of functions equibounded in $$L^1$$ norm - that is, there exists some $$M > 0$$ such that $$\|f_i\|_{L^1} < M$$.

Define the functional $$F: L^1([0, 1]) \to \mathbb R$$ by

$$F(h) = \limsup_{i \to \infty} \|f_i - h\|_{L^1}.$$

Question: Does this functional admit a minimiser? Is the minimiser unique whenever it exists?

Remarks:

What I have tried so far is to attempt to apply the direct method of the calculus of variations.

Since the $$f_i$$ are equibounded in $$L^1$$, it can be shown that $$F$$ is coercive, thus any minimising sequence is bounded in $$L^1$$ norm. In particular we have a weakly-* converging subsequence, say $$h_n \overset{*}{\to} h$$.

The result would follow if we had weak-* sequential lower semi continuity of $$F$$ at the minimiser - that is, that

$$\liminf_{n \to \infty} F(h_n) \geq F(h).$$

I could neither disprove this with a counterexample, nor prove it in generality.

Edit: As pointed out in the comments, weak-$$*$$ convergence to an $$L^1$$ function isn’t guaranteed, only convergence to a measure.

• A minimizer is not unique in general. E.g., let $f_i=(-1)^i$. Then any $h$ with $|h|\le1$ and $\int h=0$ is a minimizer. Jul 22, 2021 at 15:03
• $L^1(0,1)$ is not a dual space. How do you define weak-* convergence?
– gerw
Jul 23, 2021 at 18:15
• Oh you’re right, it needs to be defined in the sense of measures.. but then there is trouble in making sense of $F(\mu)$ for a measure $\mu$. Hmm... Jul 24, 2021 at 4:58

As it has been already noted in the comments, the minimizer doesn't need to be unique. However, it always exists. It is not terribly hard to show but it is not trivial either, so I wonder why the question attracted so few votes.

The proof consists of two independent parts. The first one is that the limit of every minimizing sequence that converges almost everywhere (or just in measure) is a minimizer and the second one is that there exists a minimizing sequence converging almost everywhere. I will use the fact that we deal with a finite measure space though it should, probably, be irrelevant. WLOG, we may assume that $$\|f_j\|_1\le 1$$ for all $$j$$.

Part 1

Assume that $$h_n$$ is a minimizing sequence and $$h$$ is its pointwise limit (or just limit in measure). Since we can assume WLOG that $$\|h_n\|_1\le 3$$ (to be a competitor, you need to perform not much worse than $$0$$, at the very least), we have $$\|h\|_1\le 3$$ as well (by Fatou). Then, by the definition of the convergence in measure, we can write $$h_n=u_n+v_n$$ where $$u_n$$ converge to $$h$$ uniformly and $$v_n$$ are supported on $$E_n$$ with $$m(E_n)\to 0$$ as $$n\to\infty$$. Also $$\|v_n\|_1\le 7$$, say.

Now, since $$v_n\in L^1$$, we can find $$\delta_n>0$$ such that for every set $$E$$ with $$m(E)<\delta_n$$, we have $$\int_E|v_n|<\frac 1n$$, say. By induction, we can now choose a subsequence $$n_k$$ such that $$\sum_{q=k+1}^\infty m(E_{n_q})<\delta_{n_k}\,.$$ Then $$\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$$ and, adding these parts to $$u_{n_k}$$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $$h_n=U_n+V_n$$ where $$U_n\to h$$ in $$L^1$$ and $$V_n$$ have disjoint supports $$G_n$$.

Since correcting a minimizing sequence by a sequence tending to $$0$$ in $$L^1$$ results in a minimizing sequence, we can just as well assume that $$h_n=h+V_n$$.

If $$\|V_n\|_1\to 0$$ (even along a subsequence), we are done. Assume now that $$\|V_n\|_1\ge \tau>0$$. Then for every function $$g$$ of $$L^1$$-norm less than $$4$$, we have $$\|g\|_1\le \max(\|g-V_n\|_1,\dots,\|g-V_{n+N-1}\|_1)$$ for all $$n$$ as soon as $$N>8/\tau$$. Indeed, since the supports $$G_n$$ of $$V_n$$ are disjoint, there is $$q\in\{0,\dots,N-1)$$ such that $$\int_{G_{n+q}}|g|\le \frac 1N\|g\|_1< \frac\tau 2$$, in which case subtracting $$V_{n+q}$$ can only drive the norm up.

Applying this to the functions $$g_j=f_j-h$$, we conclude that $$\limsup_{j\to\infty}\|f_j-h\|_1\le \limsup_{j\to\infty}\max_{0\le q\le N-1}\|f_j-h_{n+q}\|_1= \max_{0\le q\le N-1}\limsup_{j\to\infty}\|f_j-h_{n+q}\|_1$$ for every $$n$$, i.e. $$h$$ is a minimizer in this case as well.

Part 2

Let $$I$$ be the infimum of our functional. For every $$\varepsilon>0$$ consider the set $$X_\varepsilon$$ of all $$L^1$$-functions $$h$$ for which the value of the functional is at most $$I+\varepsilon$$. Clearly, it is a convex, non-empty, closed (in $$L^1$$) set and $$X_{\varepsilon'}\supset X_{\varepsilon''}$$ when $$\varepsilon'\ge\varepsilon''$$.

Fix some strictly convex non-negative function $$\Phi(t)\le |t|$$. To simplify the technicalities, I'll choose it by the conditions $$\Phi(0)=\Phi'(0)=1$$, $$\Phi''(t)=\frac 2{\pi(1+|t|^2)}$$ but pretty much any other choice will work as well.

Let $$J_\varepsilon=\inf_{h\in X_\varepsilon}\int\Phi(h)\,.$$ Clearly, $$J_\varepsilon$$ is a bounded non-increasing function on $$(0,1)$$, say. Let $$J$$ be its limit at $$0+$$. Passing to an appropriate decreasing sequence $$\varepsilon_n\to 0+$$ and re-enumerating $$X_n=X_{\varepsilon_n}, J_n=J_{\varepsilon_n}$$, we can assume that $$J_n\ge J-2^{-5n}$$ We will choose a representative $$h_n$$ of $$X_n$$ for which $$\int\Phi(h_n)$$ is not more than $$2^{-5n}$$ above $$J$$ and, thereby, not more than $$2\cdot 2^{-5n}$$ above its infimum over $$X_n$$. Then for $$m>n$$, we have $$h_{n,m}=\frac{h_n+h_m}2\in X_n$$ (convexity of $$X_n$$ plus inclusion $$X_n\supset X_m$$) and $$\int\Phi(h_{n,m})\le \int\frac 12(\Phi(h_n)+\Phi(h_m))-\int\frac{(h_n-h_m)^2}{4\pi (1+|h_n|^{2}+|h_m|^{2})}$$ (second order Taylor with the remainder in the Lagrange form), whence $$\int\frac{(h_n-h_m)^2}{4\pi(1+|h_n|^2+|h_m|^2)}\le 2\cdot 2^{-5n}$$ regardless of $$m>n$$ (otherwise we would go below $$J+2^{-5n}-2\cdot 2^{-5n}=J-2^{-5n}$$, which is below the infimum over $$X_n$$). It remains to note that if $$|h_n|\le 2^n$$ and $$|h_n-h_m|>2^{-n}$$, then the integrand is at least $$\rm{const}\, 2^{-4n}$$, so we get $$m(\{|h_n|\le 2^n, |h_n-h_m|>2^{-n}\})\le \rm{Const}\,2^{-n}$$ for all $$m>n$$ from where it follows at once that $$h_n(x)$$ is a Cauchy sequence for almost all $$x$$ (recall that $$\|h_n\|_1\le 3$$, so the first condition excludes only a set of measure $$3\cdot 2^{-n}$$, then use Borel-Cantelli).

That's it. Feel free to ask questions if anything is unclear :-)

• Wow, this looks impressive. This part was a bit confusing to me - “ Then $\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$ and, adding these parts to $u_{n_k}$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $h_n=U_n+V_n$ where $U_n\to h$ in $L^1$ and $V_n$ have disjoint supports $G_n$.” Do you mean to add $v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$ to $u_{n_k}$? I am not sure how to get the desired representation from this. Aug 2, 2021 at 4:04
• @NateRiver Exactly as you said: $U_k=u_{n_k}+v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$, $V_k=v_{n_k}\chi_{E_{n_k}\setminus \cup_{q>k}E_{n_q}}=v_{n_k}\chi_{G_k}$. Aug 2, 2021 at 7:45
• @NateRiver I just tried to solve the problem in $L^2$ first. There weak convergence is guaranteed, but doesn't seem to drop the value of the functional immediately, so I needed the norm convergence of a minimizing sequence. Fortunately, if one has a nested sequence of bounded closed convex sets in a Hilbert space, the smallest norm elements of the sets converge in norm (the proof is the same as above just using the parallelogram identity). I tried to mimic that idea in $L^1$ (which required strict convexity of something) and got the a.e. convergence this way, which turned out to be sufficient. Aug 2, 2021 at 13:33
• An alternative proof for Part 2 (existence of a minimizing sequence converging a.e.) Let $h_n$ be any minimizing sequence for $F$. As observed, it is bounded in $L_1$, so by the Komlós Theorem, up to extracting a subsequence, it is a.e. converging in Cesaro sense to some $h\in L^1$. Since $F$ is a convex functional, the sequence of the Cesaro means is still a minimizing sequence. Aug 2, 2021 at 13:38
• Of course, and I wouldn't be surprised if you answer (which I'm still reading) contains as a byproduct an alternative proof of that Komlós theorem (which is a 10 page paper: link.springer.com/article/10.1007%2FBF02020976 ). Aug 2, 2021 at 13:51


# How to properly represent a matrix function.

Given the function $$f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$$, what is the correct way to represent the matrix function in respect to the standard basis?

With the representation theorem, I would write the matrix in columns as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\\ -1 & h & 3\end{bmatrix}$$ But in my textbook it is written in rows as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & -1\\ 0 & 1 & h\\ 1 & 1 & 3\end{bmatrix}$$

What is the difference between them?

The correct way to represent the function $$f_h$$ in matrix form depends on the convention that you want to use to represent it. Let $$(x,y,z) \in \mathbb{R}^3$$. The matrix which you computed is useful for expressing $$f_h$$ as $$f_h(x,y,z) = \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 1 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix}.$$ We can verify this by expanding the matrix product: \begin{align} \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 2 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix} &= \begin{bmatrix} x \cdot 1 + y \cdot 0 + z \cdot (-1) \\ x \cdot 0 + y \cdot 1 + z \cdot h \\ x \cdot 1 + y \cdot 1 + z \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} x - z & y + hz & x + y + 3z \end{bmatrix}. \end{align} The result is a $$1 \times 3$$ matrix, which we can interpret as a row vector in $$\mathbb{R}^3$$. The matrix written in your textbook is useful for the following representation: $$f_h(x,y,z) = \begin{bmatrix} 1& 0 & -1 \\ 0 & 1& h\\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x-z \\ y + hz \\ x + y + 3z \end{bmatrix}.$$ The result is a $$3 \times 1$$ matrix, which we can regard as a column vector in $$\mathbb{R}^3$$.
The two representations look different in the sense that the former yields a row vector and the latter a column vector, however they are the same in the sense that they can both be regarded as lists of three real numbers, i.e. as $$(x-z,y+hz,x+y+3z) \in \mathbb{R}^3.$$ The accepted answer here sums this idea up pretty well.


# Representation of function as power series - unique?

The following example is from Calculus, 7e by James Stewart:

Example 2, Chapter 11.9 (Representations of functions as Power Series)

Find a power series representation for $$\frac{1}{x+2}$$

My solution is:

$$\frac{1}{x+2}=\frac{1}{1-\left(-x-1\right)}=\sum_{n=0}^{\infty}\left(-x-1\right)^{n}=\sum_{n=0}^{\infty}\left(-1\right)^{n}\left(x+1\right)^{n}$$

Then to find the interval of convergence

$$\left|x+1\right|<1 \Rightarrow x\in\left(-2,0\right)$$

But the given solution is different:

“In order to put this function in the form of the left side of Equation 1, $$\frac{1}{1-x}$$ , we first factor a 2 from the denominator:

$$\frac{1}{2+x}=\frac{1}{2\left(1+\frac{x}{2}\right)}=\frac{1}{2\left[1-\left(-\frac{x}{2}\right)\right]}=\frac{1}{2}\sum_{n=0}^{\infty}\left(-\frac{x}{2}\right)^{n}=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{2^{n+1}}x^{n}$$

This series converges when $$\left|-\frac{x}{2}\right|<1$$, that is, $$\left|x\right|<2$$. So the interval of convergence is $$\left(-2,2\right)$$."

So my questions are:

1. Did I make an error somewhere?
2. If not, are the two representations equivalent? Can there be more than one representation of a function as a power series?
• The representation as a power series at a given point is unique. Here, yours is at the point $-1$, while the solution gives the one at the point $0$. (By default, power series are usually taken at 0 when nothing is specified or obvious from context) – Clement C. Dec 29 '18 at 21:59

I think your method is right but you made a power series around $$x_0=-1$$
while they did it around $$x_0=0$$ which is what they asked for probably
Both solutions are correct: your series is a power series centered at $$-1$$, whereas the power series from the given solution is centered at $$0$$.


Question

# Let $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$ and $$\displaystyle { I }_{ 2 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx }$$, where $$f$$ is a continuous function and $$z$$ is any real number, $$\displaystyle \frac { { I }_{ 1 } }{ { I }_{ 2 } } =$$

A
32
B
12
C
1
D
none of these

Solution

## The correct option is B $$\displaystyle \frac { 3 }{ 2 }$$We have, $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( \left( 3-x \right) \left( 3-\left( 3-x \right) \right) \right) } dx$$   $$\displaystyle \left[ \because \int _{ a }^{ b }{ f\left( x \right)dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } \right]$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =3\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx } -\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) } dx$$$$=3{ I }_{ 2 }-{ I }_{ 1 }$$$$\displaystyle \therefore 2{ I }_{ 1 }=3{ I }_{ 2 }\Rightarrow \frac { { I }_{ 1 } }{ { I }_{ 2 } } =\frac { 3 }{ 2 }$$Mathematics

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