Short Answer

Consider an matrix A, an orthogonal $n \times n$ matrix S, and an orthogonal $m \times m$ matrix R. Compare the singular values of A with those of SAR.

A and SAR have the same singular values

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: To find the value of SAR

Given that, A is an matrix, S is an orthogonal $n \times n$ matrix and R is an orthogonal matrix.

Let, $σ$ be a singular value of SAR. Then by definition, $σ^{2}$ is an eigenvalue of. $(S A R)^{t} (S A R)$ Now we have

$\begin{array}{rcl} (S A R)^{t} (S A R) & = & R^{t} A^{t} S^{t} S A R = R^{t} A^{t} (S^{t} S) A R \\ = & R^{t} A^{t} (I_{n}) A R [Sin c e S i s a n o r t h o g o n a l m a t r i x] \\ = & R^{t} A^{t} A R \\ = & R^{- 1} A^{t} A R [Sin c e R i s a n o r t h o g o n a l m a t r i x, s o R^{t} = R^{- 1}] \end{array}$

Step 2: Compare the singular values of A with those of SAR

Hence $σ^{2}$ is an eigen value of $R^{- 1} A^{t} A R$ Clearly, the matrix $R^{- 1} A^{t} A R$ is similar to the matrix.

Hence $R^{- 1} A^{t} A R$ and $A^{t} A$ have same eigen values .Therefore, $σ^{2}$ is an eigen value of Thus is a singular value of A.

Conversely, let $σ$ be a singular value of A. Then by definition, $σ^{2}$ is an eigenvalue of $A^{t} A .$ Since, the matrix $R^{- 1} A^{t} A R$ is similar to the matrix $A^{t} A$ , therefor $R^{- 1} A^{t} A R a n d A^{t} A$ have same eigenvalues. Thus $, σ^{2}$ is an eigenvalue of $R^{- 1} A^{t} A R .$

Now as, $R^{- 1} A^{t} A R = (S A R)^{t} (S A R),$ therefore $σ$ is a singular value of SAR.

Thus A and SAR have the same singular values.

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Linear Algebra With Applications

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Short Answer

Consider an matrix A, an orthogonal $n \times n$ matrix S, and an orthogonal $m \times m$ matrix R. Compare the singular values of A with those of SAR.

Step by Step Solution

Step 1: To find the value of SAR

Step 2: Compare the singular values of A with those of SAR

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Linear Algebra With Applications

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Short Answer

Consider an matrix A, an orthogonal n×nmatrix S, and an orthogonal m×mmatrix R. Compare the singular values of A with those of SAR.

Step by Step Solution

Step 1: To find the value of SAR

Step 2: Compare the singular values of A with those of SAR

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Consider an matrix A, an orthogonal $n \times n$ matrix S, and an orthogonal $m \times m$ matrix R. Compare the singular values of A with those of SAR.