Positivity Criteria Generalizing the Leading Principal Minors Criterion

Positivity Criteria Generalizing the Leading Principal Minors Criterion An n×n Hermitian matrix is positive definite if and only if all leading principal minors Δ1, . . . ,Δn are positive. We show that certain sums δ l of l × l principal minors can be used instead of Δ l in this criterion. We describe all suitable sums δ l for 3 × 3 Hermitian matrices. For an n×n Hermitian matrix A partitioned into blocks A ij with square diagonal blocks, we prove that A is positive definite if and only if the following numbers σ l are positive: σ l is the sum of all l × l principal minors that contain the leading block submatrix [A ij ] k −1 i,j =1 (if k > 1) and that are contained in [A ij ] k i,j =1, where k is the index of the block A kk containing the (l, l) diagonal entry of A. We also prove that σ l can be used instead of Δ l in other inertia problems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Positivity Criteria Generalizing the Leading Principal Minors Criterion

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Publisher
Springer Journals
Copyright
Copyright © 2006 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-2013-2
Publisher site
See Article on Publisher Site

Abstract

An n×n Hermitian matrix is positive definite if and only if all leading principal minors Δ1, . . . ,Δn are positive. We show that certain sums δ l of l × l principal minors can be used instead of Δ l in this criterion. We describe all suitable sums δ l for 3 × 3 Hermitian matrices. For an n×n Hermitian matrix A partitioned into blocks A ij with square diagonal blocks, we prove that A is positive definite if and only if the following numbers σ l are positive: σ l is the sum of all l × l principal minors that contain the leading block submatrix [A ij ] k −1 i,j =1 (if k > 1) and that are contained in [A ij ] k i,j =1, where k is the index of the block A kk containing the (l, l) diagonal entry of A. We also prove that σ l can be used instead of Δ l in other inertia problems.

Journal

PositivitySpringer Journals

Published: Oct 13, 2006

References

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