Show that the spectral radius satisfies $\rho(AB)\le \rho(A)\rho(B)$ for commuting matrices

Question

For any square matrix $C$ with real entries, denote by $\rho(C)$ its spectral radius, i.e. the maximum magnitude of its eigenvalues. For symmetric matrices $A$ and $B$ with $AB=BA$ show that $$\rho(AB)\le \rho(A)\rho(B)$$

I think simultaneous diagonalization of $A$ and $B$ is to be used here, but couldn't find my way out.

Also will the proposition hold if the condition of symmetry is dropped?

Answer 1

Symmetry is not required. Spectral radius formula says $\rho (A)= \lim \|A^{n}\|^{1/n}$. If $AB=BA$ then $\|(AB)^{n}\|=\|A^nB^n\| \leq \|A^n\|\|B^n\|$. Take $n$-th roots and take the limit.

Answer 2

Since both matricea are simultaneously diagonalizable you can also simultaneously diagonlize $AB$ and find a link between its eigenvalues and the eigenvalues of $A$ and $B$.

A further hint: Calculate $P^{-1}AP\cdot P^{-1}BP$

Answer 3

If $A$ and $B$ are symmetric with $AB = BA$, you have the following even simpler proof : $$\|AB\|_2 = \sqrt{\rho((AB)^TAB)} = \sqrt{\rho((AB)^2)} = \rho(AB)$$ so that by sub-multiplicativity of the spectral norm, $\rho(AB) \leq \|A\|_2 \cdot \|B\|_2$ and the result follows by symmetry of $A$ and $B$ for which $\|\cdot\|_2 = \rho$.