A Linear Algebra Proof that the Inverse of a Strictly Ultrametric Matrix is a Strictly Diagonally Dominant Stieltjes Matrix

Abstract

It is well known that every $n \times n$ Stieltjes matrix has an inverse that is an $n \times n$ nonsingular symmetric matrix with nonnegative entries, and it is also easily seen that the converse of this statement fails in general to be true for $n > 2$. In the preceding paper by Martínet, Michon, and San Martín [SIAM J. Matrix Anal. Appl., 15 (1994), pp. 98–106], such a converse result is in fact shown to be true for the new class of strictly ultrametric matrices. A simpler proof of this basic result is given here, using more familiar tools from linear algebra.