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.
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The first author was partially supported by CNPq, processo 307812/2004-9. The second author was partially supported by FAPESP (São Paulo), processo 05/59407-6.
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Futorny, V., Sergeichuk, V. & Zharko, N. Positivity Criteria Generalizing the Leading Principal Minors Criterion. Positivity 11, 191–199 (2007). https://doi.org/10.1007/s11117-006-2013-2
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DOI: https://doi.org/10.1007/s11117-006-2013-2