# norm of (sub)stochastic matrix

Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.

• Which norm are you interested in? If talking about the norm induced by the infinity norm over the state space ($\|x\| = \max\limits_{i}|x_i|$) then in general you can only say that for the sub-stochastic matrix $A$ you have $\|A\| := \sup\limits_{\|x\| =1}\|Ax\| \leq 1$. Further bounds depend on the structure of $A$.
– SBF
Commented Sep 4, 2012 at 12:00
• I mean spectral norm.
– hayu
Commented Sep 4, 2012 at 12:18

A useful and easy to compute bound is given by the reasonably well-known relation (see e.g., this Wikipedia section) \begin{equation*} \|A\| \le \sqrt{\|A\|_\infty \|A\|_1} \end{equation*} between the spectral norm, and the induced $1$ and $\infty$ norms of an arbitrary matrix $A$.
Corollary: If $A$ is elementwise nonnegative and row-stochastic, then $\|A\| \le \sqrt{\|A\|_1}$.