Criteria for sign regularity of sets of matrices. (English) Zbl 0534.15017
A real \(n\times n\)-matrix A is strictly sign-regular iff all its minors are non-zero, and all \(k\times k\) minors have the same sign \(\sigma_ k(A)\), \(k=1(1)n\). Let \(A^*=DAD\), where \(D=diag(1,-1,...,(-1)^{n+1})\), and define \(A\leq^*B\) iff \(A^*\leq B^*\) entrywise. The matrix interval (relative to \(\leq^*)\) with smallest element \b{A} and largest element \(\bar A\) is \([A]=[\underline A,\bar A]\). The author proves that [A] is strictly sign-regular iff \b{A} and \(\bar A\) are strictly sign regular and \(\sigma_ k(\underline A)=\sigma_ k(\bar A)\), \(k=1(1)n\). He proves several related results, and conjectures that [A] is nonsingular and totally nonnegative iff \b{A} and \(\bar A\) are nonsingular and totally nonnegative.
Reviewer: D.Carlson
MSC:
15B57 | Hermitian, skew-Hermitian, and related matrices |
65G30 | Interval and finite arithmetic |
15B48 | Positive matrices and their generalizations; cones of matrices |
15A45 | Miscellaneous inequalities involving matrices |
Keywords:
”checkerboard” partial ordering; sign-regular matrices; totally nonnegative matrices; interval matrices; interval arithmetic; matrix intervalReferences:
[1] | Gantmacher, F. R.; Krein, M. G., Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme (1960), Akademie-Verlag: Akademie-Verlag Berlin · Zbl 0088.25103 |
[2] | Garloff, J., Untersuchungen zur Intervallinterpolation, (Dissertation (1980), Institut für Angewandte Mathematik, Universität Freiburg i. Br), Freiburger Intervall-Beriche 80/5 · Zbl 0439.41002 |
[3] | Garloff, J., Totally nonnegative interval matrices, (Nickel, K., Interval Mathematics 1980 (1980), Academic: Academic New York) · Zbl 0537.15012 |
[4] | Karlin, S., Total Positivity, Vol. I (1968), Stanford U.P: Stanford U.P Stanford · Zbl 0219.47030 |
[5] | Kuttler, J. R., A fourth-order finite-difference approximation for the fixed membrane eigenproblem, Math. Comp., 25, 237-256 (1971) · Zbl 0243.65065 |
[6] | Metelmann, K., Ein Kriterium für den Nachweis der Totalnichtnegativität von Bandmatrizen, Linear Algebra Appl., 7, 163-171 (1973) · Zbl 0254.15013 |
[7] | Roth, W. E., On the characteristic polynomial of the product of two matrices, Proc. Amer. Math. Soc., 5, 1-3 (1954) · Zbl 0055.00905 |
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