Q2E Let 聽be an orthogonal 聽2X2 mat... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q2E (page 1)

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

Short Answer

Expert verified

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

Step by step solution

of 2: Given information

It is given that A is an orthogonal 2x2 matrix.

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 ${||L (v_{1})||}^{2} = ({Av}_{1}) x ({Av}_{1}) = v_{1}^{T} A^{T} {Av}_{1}$

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