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Q7.4-23E

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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

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.

23. Let \(U = \left( {{u_1}...{u_m}} \right)\) and \(V = \left( {{v_1}...{v_n}} \right)\) where the \({{\bf{u}}_i}\) and \({{\bf{v}}_i}\) are in Theorem 10. Show that \(A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T\).

It is verified that \(A = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the matrix 

It is given that, \(U = \left( {{{\bf{u}}_{\bf{1}}}...{{\bf{u}}_{\bf{m}}}} \right)\) and \(V = \left( {{{\bf{v}}_{\bf{1}}}...{{\bf{v}}_{\bf{n}}}} \right)\)

Therefore, \({V^T} = \left( {\begin{array}{*{20}{c}}{{\bf{v}}_1^T}\\{{\bf{v}}_n^T}\end{array}} \right)\)

Step 2: Compute the matrix A from SVD:

From theorem 10, \(U\sum = \left\{ {\left( {{\sigma _1}{{\bf{u}}_1}...{\sigma _r}{{\bf{u}}_r}} \right)} \right\}\) where \(r\) is the rank of the matrix \(A\), thus find \(A = U\sum {V^T}\).

\(\begin{array}{c}A = U\sum {V^T}\\ = \left\{ {\left( {{\sigma _1}{{\bf{u}}_1}...{\sigma _r}{{\bf{u}}_r}} \right)} \right\}\left( {\begin{array}{*{20}{c}}{{\bf{v}}_1^T}\\{{\bf{v}}_n^T}\end{array}} \right)\\ = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}\end{array}\)

Hence, \(A = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}\).

Most popular questions for Math Textbooks

Classify the quadratic forms in Exercises 9–18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct \(P\) using the methods of Section 7.1.

15. \( - {\bf{3}}x_{\bf{1}}^{\bf{2}} - {\bf{7}}x_{\bf{2}}^{\bf{2}} - {\bf{10}}x_{\bf{3}}^{\bf{2}} - {\bf{10}}x_{\bf{4}}^{\bf{2}} + {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{4}}{x_{\bf{1}}}{x_{\bf{3}}} + {x_{\bf{1}}}{x_{\bf{4}}} + {\bf{6}}{x_{\bf{3}}}{x_{\bf{4}}}\)

Question: In Exercises 17-22, determine which sets of vectors are orthonormal. If a set is only orthogonal, normalize the vectors to produce an orthonormal set.

18. \(\left( {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right),{\rm{ }}\left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\end{array}} \right)\)

Question: Mark Each statement True or False. Justify each answer. In each part, A represents an \(n \times n\) matrix.

  1. If A is orthogonally diagonizable, then A is symmetric.
  2. If A is an orthogonal matrix, then A is symmetric.
  3. If A is an orthogonal matrix, then \(\left\| {A{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
  4. The principal axes of a quadratic from \({{\bf{x}}^T}A{\bf{x}}\) can be the columns of any matrix P that diagonalizes A.
  5. If P is an \(n \times n\) matrix with orthogonal columns, then \({P^T} = {P^{ - {\bf{1}}}}\).
  6. If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
  7. If \({{\bf{x}}^T}A{\bf{x}} > {\bf{0}}\) for some x, then the quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is positive definite.
  8. By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
  9. The largest value of a quadratic form \({{\bf{x}}^T}A{\bf{x}}\), for \(\left\| {\bf{x}} \right\| = {\bf{1}}\) is the largest entery on the diagonal A.
  10. The maximum value of a positive definite quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is the greatest eigenvalue of A.
  11. A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable \({\bf{x}} = P{\bf{u}}\), for some orthogonal matrix P.
  12. An indefinite quadratic form is one whose eigenvalues are not definite.
  13. If P is an \(n \times n\) orthogonal matrix, then the change of variable \({\bf{x}} = P{\bf{u}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form whose matrix is \({P^{ - {\bf{1}}}}AP\).
  14. If U is \(m \times n\) with orthogonal columns, then \(U{U^T}{\bf{x}}\) is the orthogonal projection of x onto ColU.
  15. If B is \(m \times n\) and x is a unit vector in \({\mathbb{R}^n}\), then \(\left\| {B{\bf{x}}} \right\| \le {\sigma _{\bf{1}}}\), where \({\sigma _{\bf{1}}}\) is the first singular value of B.
  16. A singular value decomposition of an \(m \times n\) matrix B can be written as \(B = P\Sigma Q\), where P is an \(m \times n\) orthogonal matrix and \(\Sigma \) is an \(m \times n\) diagonal matrix.
  17. If A is \(n \times n\), then A and \({A^T}A\) have the same singular values.

Suppose Aand B are orthogonally diagonalizable and \(AB = BA\). Explain why \(AB\) is also orthogonally diagonalizable.

Question: Find the principal components of the data for Exercise 2.
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