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.