Eigenvectors of Stochastic Matrices

We say that a matrix \(\mathbf{P} \in \mathbb{R}^{n\times n}\) is column-stochastic if \(p_{ij} \geq 0\) for all \(i\) and \(j\) and \(\sum_{i=1}^n p_{ij} = 1\) for all \(j\).

Let \(\mathbf{P}\) be column-stochastic. Prove that there exists a vector \(\mathbf{v}\) such that \(\mathbf{v} = \mathbf{P}\mathbf{v}\).

Hint: interpret column-stochasticity as a statement about an eigenvector of the matrix \(\mathbf{P}\). Recall that matrices can have both left- and right-eigenvectors.