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1. (i) Show that if A is an n x m matrix and if B is an m x n matrix, then A 0 is an eigenvalue of the n x n matrix AB if and only if A is an eigenvalue of the m x m matrix BA. Show that if m = n then the conclusion is true even for A= 0. (ii) Let M be an mxn matrix (m < n) over R. Show that at least one eigenvalue of the n x n matrix MTM is equal to 0. Show that the eigenvalues of the m xm matrix MMT are also eigenvalues of MT M. %3D

1. (i) Show that if A is an n x m matrix and if B is an m x n matrix, then A 0 is an eigenvalue of the n x n matrix AB if and only if A is an eigenvalue of the m x m matrix BA. Show that if m = n then the conclusion is true even for A= 0. (ii) Let M be an mxn matrix (m < n) over R. Show that at least one eigenvalue of the n x n matrix MTM is equal to 0. Show that the eigenvalues of the m xm matrix MMT are also eigenvalues of MT M. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. (i) Show that if A is an n × m matrix and if B is an m × n matrix, then A #0 is an eigenvalue of the n x n matrix AB if and only if ) is an eigenvalue of the m x m matrix BA. Show that if m = n then the conclusion is true even for )= 0. (ii) Let M be an mxn matrix (m < n) over R. Show that at least one eigenvalue of the n x n matrix MTM is equal to 0. Show that the eigenvalues of the m x m matrix MMT are also eigenvalues of MT M.
Transcribed Image Text:1. (i) Show that if A is an n × m matrix and if B is an m × n matrix, then A #0 is an eigenvalue of the n x n matrix AB if and only if ) is an eigenvalue of the m x m matrix BA. Show that if m = n then the conclusion is true even for )= 0. (ii) Let M be an mxn matrix (m < n) over R. Show that at least one eigenvalue of the n x n matrix MTM is equal to 0. Show that the eigenvalues of the m x m matrix MMT are also eigenvalues of MT M.
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