For n × n matrices A and B do AB and BA always have the...

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Math 307: Problems for section 4.6 November 30, 2012 1. For n × n matrices A and B do AB and BA always have the same eigenvalues? Use MAT- LAB/Octave to guess an answer and then verify your guess in the special case that one of the matrices, say A , is invertible. Do they have the same singular values? What happens when A is n × m and B is m × n matrices with n 6 = m ? Guess the answer using MATLAB/Octave.
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which shows that AB and BA have the same characteristic polynomial, and therefore the same eigen- values. When A is n × m and B is m × n matrices with n 6 = m then AB is n × n and BA is m × m so the number of eigenvalues is different. Let’s see what happens with a random matrix. > A=rand(2,4); > B=rand(4,2);

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