Question

I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$

The argument I was using to show this is that for any two square matrices $A$ and $B,$ it is always the case that $\|AB\|_2 = \|BA\|_2.$ I thought I read that this equality does hold in a reputable source, but I don't have access to it right now and I was unsuccessful in reproducing a proof.

I know the eigenvalues of $AB$ and $BA$ are the same modulo possibly having different numbers of zeros, so I'm worried that I might have ``remembered'' something that isn't true!

Any references/counterexamples for either question?

Answer 1

I suppose that your norm is either the Schur-Frobenius $\|F\|=({\rm Tr}FF^T)^{1/2}$ or that subordinated to the Euclidan norm $\|F\|=\rho(FF^T)$. In either case, the equality $\|AB\|=\|BA\|$ follows from the equality of the spectra of $ABBA$ and $BAAB$ (mind that $A$ and $B$ are symmetric). This is true because both matrices are similar to $A^2B^2$ (mind that $A$ and $B$ are non-singular).

If you drop your assumption that $A,B$ are PSD, then the equality does not hold in general. A simple counter-example is given by a pair $(A,B)$ such that $AB=0_n$ but $BA\neq 0_n$.

Answer 2

Going back to the original motivation, the following observations might still be useful.

1. Denis already mentioned this: $A^{1/2}BA^{1/2} \sim A^{1/2}A^{1/2}BA^{1/2}A^{-1/2}$ for strictly positive $A$. This implies that $\sigma(A^{1/2}BA^{1/2})=\sigma(AB)$.

2. If $A$ and $B$ are arbitrary (square) matrices such that their product $AB$ is normal, then one can show that $\sigma(AB) \prec_w \sigma(BA)$ (where $\prec_w$ denotes weak majorization); equivalently, for any unitarily invariant norm, we have $\|AB\| \le \|BA\|$.

3. If $\|AB\| \le 1$ for positive definite $A$ and $B$, then $\|A^{1/2}B^{1/2}\| \le 1$.