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Linear Algebra and its Applications
Found in: Page 395
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Question 7: Prove that an \(n \times n\) A is positive definite if and only if A admits a Cholesky factorization, namely, \(A = {R^T}R\) for some invertible upper triangular matrix R whose diagonal entries are all positive. (Hint; Use a QR factorization and Exercise 26 in Section 7.2.)

It is proved that a \(n \times n\) matrix A is positive definite if and only if A admits a Cholesky factorization.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: QR factorization

Theorem 12 in section 6.4 states that when \(A\) is a \(m \times n\) matrix that is linearly independent columns, then \(A\) may be factored as \(A = QR\), with \(Q\) is an \(m \times n\) matrix wherein columns provide an orthonormal basis for \({\mathop{\rm Col}\nolimits} A\), and \(R\) is an \(n \times n\) upper triangular invertible matrix which has positive entries on its diagonal.

Step 2: Show that \(n \times n\) A is positive definite if and only if A admits a Cholesky factorization

Exercise 25 in section 7.2 states that when \(B\) is a \(m \times n\) matrix then \({B^T}B\) is positive semidefinite and when \(B\) is a \(n \times n\) matrix then \({B^T}B\) is positive definite.

Exercise 26 in section 7.2 states that when an \(n \times n\) A matrix is positive definite, then there is a positive definite matrix \(B\) such that \(A = {B^T}B\).

When \(A\) permits a Cholesky factorization, namely \(A = {R^T}R\), with \(R\) is the upper triangular with positive diagonal entries then \(\det R = {r_{11}}{r_{22}} \cdots {r_{nn}} > 0\) and R is invertible.

Thus, A is a positive definite according to Exercise 25 in Section 7.2.

Assume that \(A\) is positive definite. Then \(A = {B^T}B\) for any positive definite matrix B, according to Exercise 26 in section 7.2. \(B\) is invertible and 0 is not an eigenvalue of \(B\) because its eigenvalues are positive.

Therefore, the columns of \(B\) are linearly independent. According to theorem 12 in section 6.4, \(B = QR\) for some \(n \times n\) matrix \(Q\) has orthonormal columns, and also some upper triangular matrix \(R\) has positive diagonal entries.

So, \({Q^T}Q = I\) because \(Q\) is a square matrix,

\(\begin{array}{c}A = {B^T}B\\ = {\left( {QR} \right)^T}\left( {QR} \right)\\ = {R^T}{Q^T}QR\\ = {R^T}R\end{array}\)

Therefore, \(R\) possess the necessary properties.

Hence, it is proved that a \(n \times n\) matrix A is positive definite if and only if A admits a Cholesky factorization.

Most popular questions for Math Textbooks

Orhtogonally diagonalize the matrices in Exercises 37-40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and for each eigenvalue \(\lambda \), find an orthogonal basis for \({\bf{Nul}}\left( {A - \lambda I} \right)\), as in Examples 2 and 3.

37. \(\left( {\begin{aligned}{{}}{\bf{6}}&{\bf{2}}&{\bf{9}}&{ - {\bf{6}}}\\{\bf{2}}&{\bf{6}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{9}}&{ - {\bf{6}}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{\bf{9}}&{\bf{2}}&{\bf{6}}\end{aligned}} \right)\)

Find the matrix of the quadratic form. Assume x is in \({\mathbb{R}^2}\) .

a. \(3x_1^2 - 4{x_1}{x_2} + 5x_2^2\) b. \(3x_1^2 + 2{x_1}{x_2}\)

10.Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

10. \(\left( {\begin{aligned}{{}}{1/3}&{\,\,2/3}&{\,\,2/3}\\{2/3}&{\,\,1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\)

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

a. An \(n \times n\) matrix that is orthogonally diagonalizable must be symmetric.

b. If \({A^T} = A\) and if vectors \({\rm{u}}\) and \({\rm{v}}\) satisfy \(A{\rm{u}} = {\rm{3u}}\) and \(A{\rm{v}} = {\rm{3v}}\), then \({\rm{u}} \cdot {\rm{v}} = {\rm{0}}\).

c. An \(n \times n\) symmetric matrix has n distinct real eigenvalues.

d. For a nonzero \({\rm{v}}\) in \({\mathbb{R}^n}\) , the matrix \({\rm{v}}{{\rm{v}}^T}\) is called a projection matrix.

25. Let \({\bf{T:}}{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) be a linear transformation. Describe how to find a basis \(B\) for \({\mathbb{R}^n}\) and a basis \(C\) for \({\mathbb{R}^m}\) such that the matrix for \(T\) relative to \(B\) and \(C\) is an \(m \times n\) “diagonal” matrix.
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