TITLE: Eigenvalues of a product of matrices, involving Moore-Penrose pseudo inverse
QUESTION [6 upvotes]: I have recently came across the following question in my linear algebra research, which I suspect has a basic straightforward answer:

Suppose we have a real square matrix $ A $, and a certain real square matrix $ B $, which is not necessarily invertible. If $ B $ is invertible, then the matrices $ A $ and $ BAB^{-1} $ and $ B^{-1}AB $ have the same eigenvalues due to similarity. However, if $ B $ is not invertible, then what about the matrices $ BAB^{\dagger} $ and $ B^{\dagger}AB $  ($ \dagger $ denotes Moore-Penrose pseudoinverse) Do they share eigenvalues with $ A $? Is there an analogy here?

I really have no ability to answer this question, I am not an expert on the pseudo inverse of matrices, so I am hoping someone here can provide the answer, which I suspect exists. I thank all helpers.

REPLY [2 votes]: If $B$ is not invertible, there is no point seeking similarity. However, the quest for analogy for the Moore-Penrose inverse in the general case of rectangular matrices makes sense. One might need to examine whether similarity would apply for a full-rank matrix $B$. Which by conjecture seems to be the case.