Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

The singular values of A are $σ_{1} = 1 and σ_{2} = 1$ are derived using the theorem 8.3.2

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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 is an orthogonal 2x2 matrix.

Step 2 of 2: Find the singular value

Let us have $v_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}] and v_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}], where \{v_{1}, v_{2}\}$ forms an orthonormal basis of .

Here, the unit circle consists of the vectors of the form $x = \cos (t) v_{1} + \sin (t) v_{1}$ and the image of the unit circle consists of the vectors of the form

$L (x) = \cos (t) L (v_{1}) + \sin (t) L (v_{2})$

Therefore, the image is the ellipse whose semi-major and semi-minor axes are $L (v_{1}) and L (v_{2})$ and respectively.

The length of the axes are ${||L (v_{1})||}^{2} = ({Av}_{1}) x ({Av}_{1}) = v_{1}^{T} A^{T} {Av}_{1}$

, $= v_{1}^{T} l_{2} v_{1}, where A is an orthogonal matrix = v_{1}^{T} v_{1} = 1 as v_{1} is an unit vector ||L {(v_{2})}^{2}|| = ({Av}_{2}) x ({Av}_{2}) = v_{1}^{T} A^{T} {Av}_{2}, = v_{1}^{T} l_{2} v_{1} where A is an orthogonal matrix = v_{1}^{T} v_{1} = 1 as v_{2} is an unit vector$

Thus $||L {(v_{2})}^{2}|| = 1, ||L {(v_{2})}^{2}|| = 1$

Thus, the singular values of A are $σ_{1} = 1 and σ_{2} = 1$ .

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

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Linear Algebra With Applications

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Short Answer

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

Step by Step Solution

Step 1 of 2: Given information

Step 2 of 2: Find the singular value

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Linear Algebra With Applications

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Short Answer

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

Step by Step Solution

Step 1 of 2: Given information

Step 2 of 2: Find the singular value

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