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."