Let $A=(a_{i,j})_{i,j=1}^n$ be a quadratic matrix with entries in $\mathbb{C}$. We define the i-th Gerschgorin circle by its center $m_i:=a_{i,i}$, which is the i-th diagonal entry of the matrix $A$, and the associated radius $r_i:=\sum_{j=1,j\neq i}^n |a_{j,i}|$, which is the sum of the entries of the i-th row without the diagonal entry.
How can I conclude from the Gerschgorin circle theorem that a stochastic matrix is invertible if all diagonal entries are larger than $\frac{1}{2}$?
For the case of a stochastic matrix with rows summing to $1$, if the diagonal entries are larger than $1/2$, then the Gerschgorin disc radii are all smaller than $1/2$. Therefore they do not contain $0$, and hence $0$ can not be an eigenvalue of the stochastic matrix, implying the matrix is invertible. If you have a stochastic matrix with columns summing to $1$, just apply the reasoning on the transpose of the matrix and you'll arrive at the same conclusion.