The set of matrices with same spectral radius

Question

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to describe this set. Or, more generally, what is the set of matrices having the same spectral radius? Even more generally, if $A$ is a Banach algebra with unit, is there a description of the subset $$A_r = \{ \ x \in A \ | \ \text{spectral radius of } x = r \ \} $$

Here is what I have. By Gelfand's theorem, $$r(x) = \lim ||x^k||^{1/k}.$$

If there was no $k$ on the above formula, then $ A_r = \{ \ x \ | \ ||x|| =r \ \}$, a ball. I have no idea how to describe this set if we put $k$ back into the equation. On the limit, is this some sort of sup norm? (Hence, in case of matrices, just take the maximum norm?). Can I describe it as some sort of coset, affine space etc? I thought about thinking of it as a dynamical system problem, turn on $k$ and see how the ball behave as $k$ goes to infinity.

Actually, the fact that $A_r$ is a ball if $k$ was not in the equation is pretty bad for me since I want to minimize a function over the set $S = \{ \ x \in A \ | \ r(x) > r_0 \ \}$. Even if there is no description of $A_r$, is there a statement about convexity of $\bigcup_{r < r_0} A_r$, or of $$S \bigcap \text{ positive matrices}. $$ Any reference on optimization over matrices are also very well come.

Answer 1

First of all, for self-adjoint matrices spectral radius equals norm. So as to your last question, $S \cap M_n^+$ is the set of positive matrices whose norm is greater than $r_0$, which is certainly not convex.

Nor is $\bigcup_{r<r_0} A_r$ convex --- even $A_0$ is not convex; e.g., take $\left[\begin{matrix} 0&1\cr 0&0\end{matrix}\right]$ and $\left[\begin{matrix} 0&0\cr 1&0\end{matrix}\right]$, both of these have spectral radius $0$ but their average is a self-adjoint matrix whose norm is $\frac{1}{2}$.

Answer 2

Since you ask about positive entries, the Perron-Frobenius theorem limits possibilities. A possible characterization is the following: $$ A_1 = \{DMD^{-1} \colon \text{$D$ diagonal with positive $D_{ii}$, $M$ with positive entries and such that $M\boldsymbol{1}=\boldsymbol{1}$}\}, $$ with $\boldsymbol{1}$ the vector of all ones.