= Irreducibly diagonally dominant matrix
A complex square <matrix> is irreducibly diagonally dominant when every row satisfies $|a_{ii}|\geq\sum_{j\ne i}|a_{ij}|$, 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.
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