 Irreducibly diagonally dominant matrix

A complex square matrix is irreducibly diagonally dominant when every row satisfies $|a_{ii}|\ge {\sum }_{j\ne i}|a_{ij}|$, at least one row is strict, and its nonzero off-diagonal entries define a strongly connected directed graph. Such a matrix is nonsingular. For if $Az=0$, choose a component of maximum modulus. The row inequality forces every out-neighbour to have that same modulus, unless the maximum is zero. Connectivity propagates this modulus to a strict row, where the row inequality is impossible. Thus the maximum is zero and $z=0$. This extends nonsingularity of a strictly diagonally dominant matrix to many finite difference matrices with strict dominance only near a boundary.