\(M\)-matrices whose inverses are totally positive
A real invertible \(n \times n\) matrix with nonpositive off-diagonal elements is an \(M\)-matrix provided \(Ax \geq 0\) implies \(x \geq 0\) for all \(x \in {\mathcal F}^n\). The author shows that if \(A\) is such a matrix, then \(A^{- 1}\) is totally positive (i.e., all minors are nonnegative) if and only if \(A\) is a tridiagonal matrix. As a kind of dual, if \(A\) is nonsingular and totally positive, then it is shown that \(A^{- 1}\) is an \(M\)-matrix if and only if \(A^{- 1}\) is tridiagonal if and only if \(\text{det} A (i |j) = 0\) whenever \(|i - j |= 2\).
scientific article; zbMATH DE number 769058