Comptes Rendus
no. 2
Algebra/Mathematical analysis
Matrix positivity preservers in fixed dimension

Presented by: the Editorial Board

Alexander Belton1; Dominique Guillot2; Apoorva Khare3; Mihai Putinar4, 5
1 Lancaster University, Lancaster, UK
2 University of Delaware, Newark, DE, USA
3 Stanford University, Stanford, CA, USA
4 University of California at Santa Barbara, CA, USA
5 Newcastle University, Newcastle upon Tyne, UK
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 143-148.

A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomial case. As consequences, we derive tight linear matrix inequalities for Hadamard powers of positive semidefinite matrices, and a sharp asymptotic bound for the matrix cube problem involving Hadamard powers.

Un résultat classique de I.J. Schoenberg caractérise les fonctions préservant la positivité lorsqu'elles sont appliquées aux entrées des matrices semi-définies positives de dimension arbitraire. Le problème analogue lorsque la dimension est fixe est beaucoup plus complexe à résoudre. Dans cette note, nous résolvons ce problème dans le cas où la fonction est un polynôme. Nous dérivons de ce résultat des inégalités exactes pour les puissances d'Hadamard d'une matrice positive et pour le problème du cube matriciel.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.11.006

Alexander Belton 1; Dominique Guillot 2; Apoorva Khare 3; Mihai Putinar 4, 5

1 Lancaster University, Lancaster, UK
2 University of Delaware, Newark, DE, USA
3 Stanford University, Stanford, CA, USA
4 University of California at Santa Barbara, CA, USA
5 Newcastle University, Newcastle upon Tyne, UK 
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Alexander Belton; Dominique Guillot; Apoorva Khare; Mihai Putinar. Matrix positivity preservers in fixed dimension. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 143-148. doi : 10.1016/j.crma.2015.11.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.006/

[1] A. Belton; D. Guillot; A. Khare; M. Putinar Matrix positivity preservers in fixed dimension. I (submitted for publication) | arXiv

[2] C. Berg; J.P.R. Christensen; P. Ressel Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Grad. Texts Math., vol. 100, Springer-Verlag, New York, 1984

[3] S. Bernstein Sur les fonctions absolument monotones, Acta Math., Volume 52 (1929) no. 1, pp. 1-66

[4] Philadelphia, PA, USA, MOS-SIAM Ser. Optim., vol. 13 (2013)

[5] S. Bochner Hilbert distances and positive definite functions, Ann. of Math., Volume 42 (1941), pp. 647-656

[6] S. Bochner Positive zonal functions on spheres, Proc. Natl. Acad. Sci. USA, Volume 40 (1954), pp. 1141-1147

[7] J. Borcea; P. Brändén The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving stability, Invent. Math., Volume 177 (2009) no. 3, pp. 541-569

[8] J. Borcea; P. Brändén The Lee–Yang and Pólya–Schur programs. II. Theory of stable polynomials and applications, Commun. Pure Appl. Math., Volume 62 (2009) no. 12, pp. 1595-1631

[9] J.P.R. Christensen; P. Ressel Functions operating on positive definite matrices and a theorem of Schoenberg, Trans. Amer. Math. Soc., Volume 243 (1978), pp. 89-95

[10] C.H. FitzGerald; R.A. Horn On fractional Hadamard powers of positive definite matrices, J. Math. Anal. Appl., Volume 61 (1977), pp. 633-642

[11] D. Guillot; A. Khare; B. Rajaratnam Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity, J. Math. Anal. Appl., Volume 425 (2015) no. 1, pp. 489-507

[12] D. Guillot; A. Khare; B. Rajaratnam Preserving positivity for rank-constrained matrices, Trans. Amer. Math. Soc. (2016) (arXiv:1406.0042), in press | DOI

[13] D. Guillot; A. Khare; B. Rajaratnam Preserving positivity for matrices with sparsity constraints, Trans. Amer. Math. Soc. (2016) (published online) | DOI

[14] D. Guillot; B. Rajaratnam Retaining positive definiteness in thresholded matrices, Linear Algebra Appl., Volume 436 (2012) no. 11, pp. 4143-4160

[15] D. Guillot; B. Rajaratnam Functions preserving positive definiteness for sparse matrices, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 627-649

[16] C.S. Herz Fonctions opérant sur les fonctions définies-positives, Ann. Inst. Fourier (Grenoble), Volume 13 (1963), pp. 161-180

[17] F. Hiai Monotonicity for entrywise functions of matrices, Linear Algebra Appl., Volume 431 (2009) no. 8, pp. 1125-1146

[18] R.A. Horn The theory of infinitely divisible matrices and kernels, Trans. Amer. Math. Soc., Volume 136 (1969), pp. 269-286

[19] J. Kahane; W. Rudin Caractérisation des fonctions qui opèrent sur les coefficients de Fourier–Stieltjes, C. R. Acad. Sci. Paris, Volume 247 (1958), pp. 773-775

[20] A. Nemirovski Advances in convex optimization: conic programming, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 413-444

[21] W. Rudin Positive definite sequences and absolutely monotonic functions, Duke Math. J., Volume 26 (1959), pp. 617-622

[22] I. Schoenberg Positive definite functions on spheres, Duke Math. J., Volume 9 (1942), pp. 96-108

[23] J. von Neumann; I. Schoenberg Fourier integrals and metric geometry, Trans. Amer. Math. Soc., Volume 50 (1941), pp. 226-251

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