Eigenvalues on the spectral radius of a non-negative matrix

Question

Consider a square and non-negative matrix $A$, of dimension $n$, such that each row of $A$ has the same sum $d$. It is known that the spectral radius of $A$ is $d$.

Now, suppose that there are $m<n$ eigenvalues of $A$ such that $|\sigma_j| = d$, $j=1,\cdots,m$, that is, $A$ has $m$ eigenvalues on its spectral circle. My question is: Do they (the eigenvalues with modulus equals $d$) always comes uniformly distributed on the spectral circle of $A$, that is, do they can always be written as? $$ \sigma_k = d \exp(2\pi ki/m), k=1,\cdots,m. $$

The Perron-Frobenius theory asserts that if the graph $\mathcal{G}(A)$, with links given by $A$, is strongly connected then the result holds and $m$ is called the index of imprimitivity of $A$. But the Perron-Frobenius theory makes no assumption regarding special properties of $A$ as I'm assuming.

So, if someone can give a concrete assertation about this result I would appreciate. Many thanks!

Answer 1

The answer is no. For instance, consider the matrix $\left(\begin{smallmatrix}A_1&0\\0&A_2\end{smallmatrix}\right)$, where $$ A = \pmatrix{0&1\\1&0}, \quad A_2 = \pmatrix{0&1&0\\0&0&1\\1&0&0} $$ Then the $5$ eigenvalues of $A$ are all on the unit circle, but not "evenly distributed".

Answer 2

WLOG $d = 1$ and $A$ is the transition matrix of a finite Markov chain. The eigenvalues of absolute value $1$ are eigenvalues of the diagonal blocks of $A$ corresponding to the irreducible classes of recurrent states. Each of these blocks has a certain period $p_j$, and its eigenvalues of are all $p_j$'th roots of unity.