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Given a column stochastic matrix $P$, I wanted to give a relation between $\|P\|$ and orthogonality of $P$.

One simple way to think about how close $P$ is to being orthogonal is $\|P^{\top}P - I\|$. Then, I simply went ahead and wrote: $$\|P\| \leq \sqrt{\|P^{\top}P - I\| + \|I\|}$$

First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?

P.S. All norms are Frobenius norms.

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    $\begingroup$ For the loss of orthogonality, you might find some interesting pointers in Section 4 here. $\endgroup$ Oct 26, 2015 at 16:28
  • $\begingroup$ @AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!! $\endgroup$ Oct 26, 2015 at 17:27

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