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Consider a matrix $A \in \mathbb{R}^{n \times n}$ with non-negative entries. What are some sufficient conditions that allow me to conclude $\left\|A\right\|_2 < 1 $?

I am interested in stochastic and sub-stochastic matrices in particular.

Leonardo Massai
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    The induced $2$-norm of every stochastic matrix is $\ge1$. It is equal to $1$ if and only if the matrix is doubly stochastic. – user1551 Jul 18 '22 at 16:45
  • And what about sub-stochastic matrices, i.e. , there is at least a row that sums up to a number that is strictly less than 1 ? – Leonardo Massai Jul 18 '22 at 16:48
  • I am not aware of any nice sufficient conditions for sub-stochastic matrices. Of course, if the matrix is strictly sub-stochastic and normal, its norm will be smaller than $1$, but very few sub-stochastic matrices are normal, so this condition is not useful. – user1551 Jul 18 '22 at 16:54
  • One probably more useful sufficient condition is that one of $A$ and $A^T$ is sub-stochastic and the other is strictly sub-stochastic. In this case, $|A|2^2\le|A|_1|A|\infty<1$. – user1551 Jul 18 '22 at 16:57
  • @user1551 Regarding your first comment: I can see that "if" holds by Birkhoff-von Neumann. Is there a quick way to see why "only if" applies? – Ben Grossmann Jul 18 '22 at 16:58
  • @BenGrossmann See this answer of mine. – user1551 Jul 18 '22 at 17:01
  • @user1551 Wonderful proof, thanks for sharing – Ben Grossmann Jul 18 '22 at 19:16

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