Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 4E from Chapter 7.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 4E
Chapter:
Problem:
Find the singular values of the matrices in Exercises
Step-by-Step Solution
Given Information
We are given with a matrix: \[ A = \left[ \begin{array} { l l } { 3 } & { 0 } \\ { 8 } & { 3 } \end{array} \right] \] We have to find the singular values of the above matrix
Step-1:
\[ A ^ { T } = \left[ \begin{array} { l l } { 3 } & { 8 } \\ { 0 } & { 3 } \end{array} \right] \] Product of A and its Transpose: \[\begin{array}{l} {A^T}A = \left[ {\begin{array}{*{20}{l}} 3&8\\ 0&3 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} 3&0\\ 8&3 \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {73}&{24}\\ {24}&9 \end{array}} \right] \end{array}\]
Step-2: Eigenvalues
Write the characteristic Equation \[ \begin{array} { l } { \left| \begin{array} { c c } { 73 - \lambda } & { 24 } \\ { 24 } & { 9 - \lambda } \end{array} \right| = 0 } \\ { ( 73 - \lambda ) ( 9 - \lambda ) - 576 = 0 } \\ { \lambda ^ { 2 } - 82 \lambda + 81 = 0 } \\ { \lambda = 1 , \lambda = 81 } \end{array} \] The eigenvalues are 81 and 1. So, singular values of A are: \[ \sqrt { 81 } , \sqrt { 1 } \] Therefore,
The singular values of A are 9 and 1
We are given with a matrix: \[ A = \left[ \begin{array} { l l } { 3 } & { 0 } \\ { 8 } & { 3 } \end{array} \right] \] We have to find the singular values of the above matrix
Step-1:
\[ A ^ { T } = \left[ \begin{array} { l l } { 3 } & { 8 } \\ { 0 } & { 3 } \end{array} \right] \] Product of A and its Transpose: \[\begin{array}{l} {A^T}A = \left[ {\begin{array}{*{20}{l}} 3&8\\ 0&3 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} 3&0\\ 8&3 \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {73}&{24}\\ {24}&9 \end{array}} \right] \end{array}\]
Step-2: Eigenvalues
Write the characteristic Equation \[ \begin{array} { l } { \left| \begin{array} { c c } { 73 - \lambda } & { 24 } \\ { 24 } & { 9 - \lambda } \end{array} \right| = 0 } \\ { ( 73 - \lambda ) ( 9 - \lambda ) - 576 = 0 } \\ { \lambda ^ { 2 } - 82 \lambda + 81 = 0 } \\ { \lambda = 1 , \lambda = 81 } \end{array} \] The eigenvalues are 81 and 1. So, singular values of A are: \[ \sqrt { 81 } , \sqrt { 1 } \] Therefore,
The singular values of A are 9 and 1