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.
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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 $\|A\| \le \sqrt{\|A\|_1}$. Note For some refinements of the first inequality mentioned above, please refer to this paper by V. Nikiforov. |
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