Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$. Let $\delta = \min _ { 1 \leqslant i \leqslant n } m _ { i , i }$. Show that $| \lambda - \delta | \leqslant 1 - \delta$. Give a geometric interpretation of this result and show that, if all diagonal terms of $M$ are strictly positive, then 1 is the only eigenvalue of $M$ with modulus 1.
Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $\delta = \min _ { 1 \leqslant i \leqslant n } m _ { i , i }$. Show that $| \lambda - \delta | \leqslant 1 - \delta$. Give a geometric interpretation of this result and show that, if all diagonal terms of $M$ are strictly positive, then 1 is the only eigenvalue of $M$ with modulus 1.