Spectrum of doubly stochastic matrices

Question

Let $M$ be a doubly stochastic matrix in which every entry is strictly positive. Prove that for any eigenvalue $\lambda$ we have $\lambda \neq 1 \implies |\lambda|< 1$ and the geometric and algebraic multiplicity of the eigenvalue $1$ are the same.

I'm sure this is trivial, but I can't see it! Thanks.

Answer 1

Hint: try to define an appropriate norm on the matrix so that it is 1 for every doubly stochastic matrix and use the usual equation Av=pv.Now apply norm on this eqn and derive the fact that |p|<1 or you can use the spectral radius formula to directly get the result.1 is the largest eigenvalue so by perron-frobenius theorem,every other eigenvalue has absolute value strictly less than 1

Answer 2

This is a consequence of the following two facts:

1. The Perron-Frobenius theorem for positive matrices.
2. If $A\ge 0$ (entrywise), then $A$ has row-sums equal to one if and only if $Ae = e$, in which $e$ denotes the all-ones vector, i.e., the spectral radius of $A$ is one.