Suppose we have an $n\times n$ right stochastic matrix $P$, i.e., $P$ is a real square matrix, with each row summing to 1. Let $I_n$ be the $n\times n$ identity. I was wondering what are the good properties of the matrix $I_n -\alpha P$ where $\alpha\in(0,1)$?
What I can see immediately is the matrix $I_n-\alpha P$ is a diagonally dominant matrix. So, it inherits all the nice properties of diagonally dominant matrices such as invertibility.
We know that $P$ has many nice properties such as spectral radius less than $1$, one eigenvalue is $1$. So, I guess $I_n -\alpha P$ also has some nice properties.