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.



norm of (sub)stochastic matrix
A useful and easy to compute bound is given by the reasonably well-known relation (see e.g., [this Wikipedia section](http://en.wikipedia.org/wiki/Matrix_norm#Examples_of_norm_equivalence))
\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
$\|A\| \le \sqrt{\|A\|\_1}$. 


**Note**
For some refinements of the first inequality mentioned above, please refer to [this paper by V. Nikiforov](http://arxiv.org/abs/math/0702722).


