Duke Mathematical Journal

Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

Julius Borcea and Petter Brändén

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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

Article information

Source
Duke Math. J., Volume 143, Number 2 (2008), 205-223.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1211819162

Digital Object Identifier
doi:10.1215/00127094-2008-018

Mathematical Reviews number (MathSciNet)
MR2420507

Zentralblatt MATH identifier
1151.15013

Subjects
Primary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
Secondary: 15A22: Matrix pencils [See also 47A56] 15A48 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 32A60: Zero sets of holomorphic functions 47B38: Operators on function spaces (general)

Citation

Borcea, Julius; Brändén, Petter. Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products. Duke Math. J. 143 (2008), no. 2, 205--223. doi:10.1215/00127094-2008-018. https://projecteuclid.org/euclid.dmj/1211819162


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References

  • R. B. Bapat, ``An interlacing theorem for tridiagonal matrices'' in Proceedings of the First Conference of the International Linear Algebra Society (Provo, Utah, 1989), Linear Algebra Appl. 150, Elsevier, Amsterdam, 1991, 331--340.
  • W. W. Barrett and C. R. Johnson, Majorization monotonicity of symmetrized Fischer products, Linear and Multilinear Algebra 34 (1993), 67--74.
  • J. Borcea and P. BräNdéN, Classification of hyperbolicity and stability preservers: The multivariate Weyl algebra case, preprint,\arxivmath/0606360v4[math.CA]
  • —, Pólya-Schur master theorems for circular domains and their boundaries, to appear in Ann. of Math. (2), preprint,\arxivmath/0607416v5[math.CV]
  • J. Borcea, P. BräNdéN, G. Csordas, and V. Vinnikov, Pólya-Schur-Lax problems: Hyperbolicity and stability preservers, workshop, Amer. Inst. of Math., Palo Alto, Calif., 2007, http://www.aimath.org/pastworkshops/polyaschurlax.html
  • P. BräNdéN, Polynomials with the half-plane property and matroid theory, preprint,\arxivmath/0605678v4[math.CO]
  • Y.-B. Choe, J. G. Oxley, A. D. Sokal, and D. G. Wagner, Homogeneous multivariate polynomials with the half-plane property, Adv. in Appl. Math. 32 (2004), 88--187.
  • K. M. Da-Fonseka [C. M. Da Fonseca], An interlacing theorem for matrices whose graph is a fixed tree (in Russian), Fundam. Prikl. Mat. 10, no. 3 (2004), 245--254.; English translation in J. Math. Sci. (N. Y.) 139 (2006), 6823--6830.
  • —, On a conjecture regarding characteristic polynomial of a matrix pair, Electron. J. Linear Algebra 13 (2005), 157--161.
  • B. A. Dubrovin, ``Matrix finite-gap operators'' (in Russian) in Current Problems in Mathematics, Vol. 23, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, 33--78.
  • S. M. Fallat, M. I. Gekhtman, and C. R. Johnson, Multiplicative principal-minor inequalities for totally nonnegative matrices, Adv. in Appl. Math. 30 (2003), 442--470.
  • S. M. Fallat and C. R. Johnson, ``Determinantal inequalities: Ancient history and recent advances'' in Algebra and Its Applications (Athens, Ohio, 1999), Contemp. Math. 259, Amer. Math. Soc., Providence, 2000, 199--212.
  • L. Gurvits, ``Hyperbolic polynomials approach to Van der Waerden/Schrijver-Valiant like conjectures: Sharper bounds, simpler proofs and algorithmic applications'' in STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (Seattle, 2006), ACM Press, New York, 2006, 417--426.
  • —, A proof of hyperbolic van der Waerden conjecture: The right generalization is the ultimate simplification, preprint,\arxivmath/0504397v3[math.CO]
  • G. H. Hardy, J. E. Littlewood, and G. PóLya, Inequalities, reprint of the 1952 ed., Cambridge Univ. Press, Cambridge, 1988.
  • J. W. Helton and V. Vinnikov, Linear matrix inequality representation of sets, Comm. Pure Appl. Math. 60 (2007), 654--674.
  • O. Holtz and B. Sturmfels, Hyperdeterminantal relations among symmetric principal minors, J. Algebra 316 (2007), 634--648.
  • L. HöRmander, Notions of Convexity, Progr. Math. 127, Birkhäuser, Boston, 1994.
  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
  • C. R. Johnson, ``The permanent-on-top conjecture: A status report'' in Current Trends in Matrix Theory (Auburn, Ala., 1986), North-Holland, New York, 1987, 167--174.
  • —, A characteristic polynomial for matrix pairs, Linear and Multilinear Algebra 25 (1989), 289--290.
  • P. D. Lax, Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math. 11 (1958), 175--194.
  • A. S. Lewis, P. A. Parrilo, and M. V. Ramana, The Lax conjecture is true, Proc. Amer. Math. Soc. 133 (2005), 2495--2499.
  • A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Math. Sci. Engrg. 143, Academic Press, New York, 1979.
  • Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Math. Soc. Monogr. (N.S.) 26, Oxford Univ. Press, New York, 2002.
  • G. T. Rublein, On a conjecture of C. Johnson, Linear and Multilinear Algebra 25 (1989), 257--267.
  • H. S. Wilf, generatingfunctionology, 3rd ed., A. K. Peters, Wellesley, Mass., 2006.