Schur complements and its applications to symmetric nonnegative and Z-matrices
| Autor: | Yizheng Fan |
|---|---|
| Jazyk: | angličtina |
| Předmět: |
General Laplacian matrix
Numerical Analysis Nonnegative matrix Algebra and Number Theory Adjacency matrix Block matrix Perron complement Hermitian matrix Combinatorics Matrix (mathematics) Weighted graph Spectrum Symmetric matrix Discrete Mathematics and Combinatorics Schur complement Geometry and Topology Laplacian matrix Centrosymmetric matrix Z-matrix Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Linear Algebra and its Applications. (1-3):289-307 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/S0024-3795(02)00327-0 |
| Popis: | In [Linear Algebra Appl. 177 (1992) 137] Smith proved that if H is a Hermitian semidefinite matrix and A is a nonsingular principal submatrix, then the eigenvalues of the Schur complement H/A interlace those of H. In this paper, we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve Perron complements. For an irreducible symmetric nonnegative matrix, we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order. We apply our results to an irreducible symmetric Z-matrix and to the adjacency matrix or the general Laplacian matrix of a connected weighted graph. The equality cases for the bounds for spectral radii or least eigenvalues are also examined. |
| Databáze: | OpenAIRE |
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