given A and B are mxn and nxm matrices. What can we say about their eigenvalues

Question

I know that the sum of the eigenvalues will be the same. But can we say something about them individually? When A is transpose(B) I'm getting something interesting

A = (1 2 0) (0 -1 2)

B = (1 0) (2 -1) (0 2)

AB has 7,3 as eigenvalues BA has 3,7,0 as eigenvalues

Do you mean for $AB$ and $BA$ to be $m\times m$ and $n\times n$? If so, please edit the title accordingly. You know that if $AB$ is $m\times n$ with $m\ne n$ then it makes no sense to talk about eigenvalues, right? — Gerry Myerson, Feb 21 '17 at 06:12

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

1

Eigenvalues aren't defined for rectangular matrices, but the singular values are closely related: The right and left singular values for rectangular matrix $M$ are the eigenvalues of $M'M$ and $MM'.$

Source: Eigenvalues of a rectangular matrix

edited Apr 13 '17 at 12:19

Community

1

answered Feb 21 '17 at 06:18

rookie

1,738

given A and B are mxn and nxm matrices. What can we say about their eigenvalues

1 Answers1