Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the singular values of $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$ .

The singular values of A are $σ_{1} = 2$ and $σ_{2} = 1$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1 of 2: Given information

It is given that $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$

By computing the eigenvalues of the square matrix, the singular values of A are found

$A^{t} A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 4 \end{matrix}]$

Then, $(x |_{2} - A^{1} A) = [\begin{matrix} x - 1 & 0 \\ 0 & x - 4 \end{matrix}]$

Step 2 of 2: Find the singular value

Let the characteristic of polynomial $A^{1} A$ be $p (x)$

Then,

$p (x) = d e t (x I_{2} - A^{1} A) = (x - 1) (x - 4) - 0 = (x - 1) (x - 4)$

Therefore, the characteristic polynomial is $(x - 1) (x - 4)$ , which implies that the roots of $p (x)$ are 1,4 .

So, the eigenvalues of $A^{t} A$ are $λ_{1} = 4$ and $λ_{2} = 1$

Thus the singular values of A are $σ_{1} = \sqrt{4} = 2$ and $σ_{2} = \sqrt{1} = 1$

Result: $σ_{1} = 2$ and $σ_{2} = 1$ are the singular values of A.

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Linear Algebra With Applications

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Short Answer

Find the singular values of $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$ .

Step by Step Solution

Step 1 of 2: Given information

Step 2 of 2: Find the singular value

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Find the singular values of $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$ .