Characterization of symmetric M-matrices as resistive inverses

• Published in: Linear algebra and its applications Vol. 430; no. 4; pp. 1336 - 1349
• Main Authors: Bendito, E., Carmona, A., Encinas, A.M., Gesto, J.M.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.02.2009; Elsevier
• Subjects: Algebra; Effective resistance; Exact sciences and technology; Green kernels; Kirchhoff index; Linear and multilinear algebra, matrix theory; M-matrices; Mathematics; Moore–Penrose inverse; Schrödinger operators; Sciences and techniques of general use; Effective resistance; Kirchhoff index; Green kernels; Moore–Penrose inverse; Schrödinger operators; M-matrices; Singular matrix; Schrôdinger operators; Moore Penrose inverse; Eigenvector; Irreducible operator; Ground state; Green function; Generalized inverse; Moore-Penrose inverse; Kernels; Non negative matrix; Effective résistance; Inverse matrix; Diagonal matrix; Symmetric matrix; Schrödinger operator; Eigenvalue problem; Stieltjes transformation; Positive operator
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.10.027

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices.