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1 June 2008 Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products
Julius Borcea, Petter Brändén
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Duke Math. J. 143(2): 205-223 (1 June 2008). DOI: 10.1215/00127094-2008-018

Abstract

For (n×n)-matrices A and B, define η(A,B)=Sdet(A[S])det(B[S']), where the summation is over all subsets of {1,,n}, S' is the complement of S, and A[S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A0 and B is Hermitian, then

(1) the polynomial η(zA,-B) has all real roots;

(2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of B if A>0; and

(3) for 1in, the roots of η(zA[{i}'],-B[{i}']) interlace those of η(zA,-B).

Assertions (1)–(3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard, Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants

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Julius Borcea. Petter Brändén. "Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products." Duke Math. J. 143 (2) 205 - 223, 1 June 2008. https://doi.org/10.1215/00127094-2008-018

Information

Published: 1 June 2008
First available in Project Euclid: 26 May 2008

zbMATH: 1151.15013
MathSciNet: MR2420507
Digital Object Identifier: 10.1215/00127094-2008-018

Subjects:
Primary: 15A15
Secondary: 15A22 , 15A48 , 30C15 , 32A60 , 47B38

Rights: Copyright © 2008 Duke University Press

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