Second-largest singular value of a row-stochastic matrix

Question

I am working on this problem. Consider a non-singular row-stochastic matrix $A$; we know that its 2-norm, $\|A\|_2$, is strictly greater than one if and only if it is not doubly stochastic.

Let $\sigma_1\geq\sigma_2\geq\dots\geq \sigma_n\geq0$ be the singular values and assume that the matrix $A$ is not doubly stochastic, so that $\|A\|_2=\sigma_1>1$ and assume that $\sigma_2\lneq\sigma_1$.

My question is about the second-largest singular value. Is it true that $\sigma_2\leq1$? Are there any conditions for this? Thank you.

Answer 1

$\sigma_2(A)$ is not always $\le1$. An obvious counterexample is a direct sum $A=X\oplus Y$ of two stochastic matrices $X$ and $Y$ such that $\sigma_1(X)>\sigma_1(Y)>1$. By perturbing such an $A$, one can also obtain a counterexample in which $A$ is positive.

I am not aware of any useful condition that guarantees that $\sigma_2(A)\le1$.