Spectrum of matrix product

Question

Let $A,B\in\mathbb{R}^{m\times n}$ be real-valued $m\times n$-matrices. What can be said about the relation of the spectra of $A^TB$ and $AB^T$? Is it true that these spectra coincide apart from zero?

Answer 1

Using singular value decomposition or by comparing characteristic polynomials you can show that the eigenvalues of $M$ and $M^T$ agree for a square matrix $M$.

Hence you can go from $A^T B$ to $B^T A$ without changing the eigenvalues and your question comes down to comparing the eigenvalues of a product $AB$ with those of $BA$. Their eigenvalues also agree, see for example this answer.