Characterization of symmetric \(M\)-matrices as resistive inverses. (English) Zbl 1165.15019
The aim of this paper is to characterize those non-negative and not necessarily diagonally dominant matrices, whose respective inverses are irreducible Stieltjes matrices, by proving that any irreducible Stieltjes matrix is a resistive inverse. This consists in identifying any irreducible Stieltjes matrix with a positive definite Schrödinger operator on a suitable connected network. Further, a generalization of the concept of effective resistance is provided, whose properties are similar with those of standard effective resistances, and a generalized formula for the inverse of the resistance matrix is derived. Finally, a special kind of generalized inverses of positive semidefinite Schrödinger operators is studied, namely the Green operators and mainly the Moore-Penrose inverses of singular, irreducible and symmetric \(M\)-matrices.
Reviewer: Mihail Voicu (Iaşi)
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A09 | Theory of matrix inversion and generalized inverses |
Keywords:
\(M\)-matrices; Schrödinger operators; Green kernels; Moore-Penrose inverse; effective resistance; Kirchhoff index; irreducible Stieltjes matricesReferences:
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