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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 8: Use Exercise 7 to show that if A is positive definite, then A has a LU factorization, \(A = LU\), where U has positive pivots on its diagonal. (The converse is true, too).

It is proved that A has a LU factorization \(A = LU\).

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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 a \(m \times n\) 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 if A is positive definite, then A has a LU factorization \(A = LU\)

Assume that \(A\) is positive definite, and suppose that Cholesky factorization of \(A = {R^T}R\) where \(R\) is an upper triangular and having positive diagonal entries. Consider that \(D\) as the diagonal matrix with diagonal entries equal to diagonal entries of \(R\). The matrix \(L = {R^T}{D^{ - 1}}\) is lower triangular wherein 1’s on its diagonal because right-multiply by a diagonal matrix scales the columns of the matrix on its left.

When \(U = DR\) then \(A = {R^T}{D^{ - 1}}DR = LU\).

Conversely, consider that A contains a LU factorization, that is \(A = LU\), where \(U\) contains positive pivots on its diagonal. Consider \(D\) as the diagonal matrix and \(\sqrt {{u_{11}}} , \ldots ,\sqrt {{u_{nn}}} \) on its diagonal. The matrix \(V = {\left( {{D^2}} \right)^{ - 1}}U\) is upper triangular wherein 1’s on its diagonal because multiply by a diagonal matrix on its left scales the rows of the matrix on its right and we obtain \(A = L{D^2}V\). As \(A\) is symmetric, then \({A^T} = {\left( {L{D^2}V} \right)^T} = {V^T}{D^2}{L^T} = A\), and \(L = {V^T}\). When \(R = DV = {D^{ - 1}}U\) then \(A = {V^T}DDV = {\left( {DV} \right)^T}\left( {DV} \right) = {R^T}R\).

Hence, it is proved that A has a LU factorization \(A = LU\).

Most popular questions for Math Textbooks

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

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

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.

In Exercises 17–24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

17. Show that if \(A\) is square, then \(\left| {{\bf{det}}A} \right|\) is the product of the singular values of \(A\).

Compute the quadratic form \({x^T}Ax\), when \(A = \left( {\begin{aligned}{{}}3&2&0\\2&2&1\\0&1&0\end{aligned}} \right)\) and

a. \(x = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right)\)

b. \(x = \left( {\begin{aligned}{{}}{ - 2}\\{ - 1}\\5\end{aligned}} \right)\)

c. \(x = \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right)\)

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.
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