Sign determinacy of M-matrix minors. (English) Zbl 0618.15013
Let A be a (singular or non-singular) M-matrix, and let G(A) denote its undirected graph. The main result is that if the longest simple cycle of G(A) has length not exceeding 3 then the sign of any minor of A depends only on G(A) and not on the magnitude of the entries of A.
Reviewer: G.P.Barker
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
References:
| [1] | Johnson, Charles R.; Olesky, D. D.; van den Driessche, P., \(M\)-matrix products having positive principal minors, Linear and Multilinear Algebra, 16, 29-38 (1984) · Zbl 0549.15012 |
| [2] | Marcus, M.; Minc, H., A Survey of Matrix Theory and Matrix Inequalities (1964), Prindle, Weber and Schmidt: Prindle, Weber and Schmidt Boston · Zbl 0126.02404 |
| [3] | Ponstein, J., Self-avoiding paths and the adjacency matrix of a graph, SIAM J. Appl. Math., 14, 600-609 (1966) · Zbl 0146.45901 |
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