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A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. (English) Zbl 0803.15020
An \(n \times n\) Stieltjes matrix \(A=(a_{ij})\) is defined as a real symmetric and positive definite matrix with \(a_{ij} \leq 0\) for all \(i \neq j\) \((1 \leq i,j \leq n)\). It is well known that every Stieltjes matrix has an inverse which is a nonsingular symmetric matrix with nonnegative entries. In this note the authors show, by giving a counterexample, that the converse of this statement fails in general to be true for \(n\geq 3\), and give a simpler proof of a partially converse theorem proved by S. Martinez, G. Michon and J. San Martan [Inverses of strictly ultrametric matrices are of Stieltjes type. SIAM J. Matrix Anal. Appl., 15, No. 1, 98-106 (1994; Zbl 0798.15030)], that a strictly ultrametric matrix \(A=(a_{ij})\) is nonsingular and its inverse \(A^{- 1} = (\alpha_{ij})\) is a strictly diagonal dominant Stieltjes matrix with the additional property: \(\alpha_{ij} = 0\) if and if \(a_{ij} = 0\), using more familiar tools from linear algebras.

15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15A09 Theory of matrix inversion and generalized inverses
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