1

I know that $AB$ and $BA$ have the same non-zero eigenvalues, where $A$ and $B$ are rectangular matrices, but my question is: Is it necessary for $AB$ and $BA$ to have the same number of non-zero eigenvalues?

In other words, suppose $AB$ is a $8*8$ matrix and $BA$ a $6 * 6$ matrix. Then can $AB$ have eigenvalues of $0, 0, 0, 0, 0, 2, 3, 4$ when $BA$ has eigenvalues of $0, 0, 2, 2, 3, 4$?

Writer's edit.

Note that the question can be answered using the generalization of the Sylvester's Determinant Identity: Sylvester's determinant identity and the generalization process: Does the following always hold? , meaning the answer to the original question, would be "True."

Richard
  • 179
  • That would be true, but what I meant was: as in the example, can eigenvalue 2 have a larger algebraic multiplicity for a matrix of a lower size? @AnyAD – Richard Jun 03 '18 at 02:41
  • @ Richard The quesrion has been asked here https://math.stackexchange.com/questions/332674/do-the-non-zero-eigenvalues-of-ab-and-ba-have-the-same-algebraic-multiplicity-f – AnyAD Jun 03 '18 at 03:15
  • @AnyAd edited my question - does that explanation help? – Richard Jun 03 '18 at 03:50

0 Answers0