Are the eigenvalues of a irreducible row-stochastic matrix all real?

Question

Let $A=[a_{ij}]\in\mathbb{R}^{n \times n}$ ($a_{ij} \ge 0$) be a irreducible row-stochastic matrix. Are all the eigenvalues of $A$ real ?

Answer 1

No. A look at Theorem 1.7 and its corollary in 'Non-negative Matrices and Markov Chains' by E. Seneta tells you how to construct such matrices with lots of non-real eigen values.

Answer 2

The answer is "yes" when $A$ is $2\times2$, because non-real eigenvalues of a real matrix must occur in a conjugate pair, but $A$ already has a real eigenvalue $1$.

When $A$ is at least $3\times3$, the answer is negative and it suffices to construct a counterexample whose size is exactly $3\times3$. In fact, if $A_3$ is a $3\times3$ counterexample, $E$ is the $n\times n$ all-one matrix and $t$ is a sufficiently small positive scalar, then $A=tE+(1-t)(A_3\oplus I_{n-3})$ is an $n\times n$ counterexample.

It remains to exhibit a $3\times3$ counterexample. The eigenvalues of the matrix below are $-1$ and $\frac1{120}(-1\pm i\sqrt{239})$: $$ A_3=\pmatrix{ \frac25&\frac15&\frac25\\ \frac12&\frac14&\frac14\\ \frac13&\frac13&\frac13}. $$