For example AB and BA have the same singular spectrum (singular values) for matrices below, is there a general characterization of cases when
- singular spectrum is the same?
- non-zero singular spectrum is the same?
$$A=\left( \begin{array}{cc} 11 & 9 \\ 9 & 11 \\ \end{array} \right)$$
$$B=\left( \begin{array}{cc} 4 & 0 \\ 0 & 1 \\ \end{array} \right)$$
$$\sigma(AB)=\sigma(BA)=\langle \sqrt{3 \sqrt{324721}+1717},\frac{160}{\sqrt{3 \sqrt{324721}+1717}}\rangle$$
A related question is when $AB$ and $BA$ have the same non-zero spectrum, the answer is "always". (proof for square matrices here)
