Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
We have solutions for your book!
See our solution for Question 20E from Chapter 7.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 20E
Chapter:
Problem:
0
Step-by-Step Solution
Given Information
We have to show that if $P$ is an orthogonal $m \times m$ matrix, then $P A$ has the same singular values as $A$
Step-1: (a)
Consider the equation\[\begin{aligned} P A & = P \left( U \Sigma V ^ { T } \right) \\ & = ( P U ) \Sigma V ^ { T } \end{aligned}\]Since $P$ and $U$ are orthogonal, so $P U$ is also orthogonal.
Since $P U$ and $V$ are orthogonal and $\Sigma$ is a diagonal matrix, so $P A = ( P U ) \Sigma V ^ { T }$ is the singular value decomposition of $P A$ . Therefore, the diagonal entries of $\Sigma$ are the singular values of $P A$ .Hence,
The matrices $A$ and $P A$ have the same singular values.
We have to show that if $P$ is an orthogonal $m \times m$ matrix, then $P A$ has the same singular values as $A$
Step-1: (a)
Consider the equation\[\begin{aligned} P A & = P \left( U \Sigma V ^ { T } \right) \\ & = ( P U ) \Sigma V ^ { T } \end{aligned}\]Since $P$ and $U$ are orthogonal, so $P U$ is also orthogonal.
Since $P U$ and $V$ are orthogonal and $\Sigma$ is a diagonal matrix, so $P A = ( P U ) \Sigma V ^ { T }$ is the singular value decomposition of $P A$ . Therefore, the diagonal entries of $\Sigma$ are the singular values of $P A$ .Hence,
The matrices $A$ and $P A$ have the same singular values.