Generalized eigenvalues of a definite hermitian matrix pair

International audience Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.

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Neural network for solving generalized eigenvalues of matrix pair

2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100) ◽

10.1109/icassp.2000.860150 ◽

2002 ◽

Cited By ~ 2

Author(s):

H.B. Yang ◽

L.C. Jiao

Keyword(s):

Neural Network ◽

Generalized Eigenvalues ◽

Matrix Pair

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Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

Advances in Calculus of Variations ◽

10.1515/acv-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Anup Biswas ◽

Prasun Roychowdhury

Keyword(s):

Eigenvalue Problem ◽

Principal Eigenvalue ◽

Unbounded Domains ◽

Elliptic Operators ◽

Maximum Principles ◽

Generalized Eigenvalues ◽

Nonlinear Elliptic ◽

Generalized Eigenvalue ◽

General Convex ◽

Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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A general inverse matrix series relation and associated polynomials – II

Mathematica Slovaca ◽

10.1515/ms-2017-0469 ◽

2021 ◽

Vol 71 (2) ◽

pp. 301-316

Author(s):

Reshma Sanjhira

Keyword(s):

Matrix Polynomial ◽

Combinatorial Identities ◽

Inverse Matrix ◽

Matrix Polynomials ◽

Special Cases ◽

Series Representations ◽

The Matrix ◽

Associated Polynomials ◽

Matrix Pair ◽

Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

Journal of High Energy Physics ◽

10.1007/jhep08(2021)032 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Bartomeu Fiol ◽

Alan Rios Fukelman

Keyword(s):

Free Energy ◽

Infinite Number ◽

Correlation Functions ◽

Hermitian Matrix ◽

Hooft Coupling ◽

Planar Limit ◽

Single Trace ◽

Hermitian Matrix Model ◽

Combinatorial Expression ◽

Do So

Abstract We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of $$ \mathcal{N} $$ N = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

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A Schur-Cohn theorem for matrix polynomials

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004806 ◽

1990 ◽

Vol 33 (3) ◽

pp. 337-366 ◽

Cited By ~ 6

Author(s):

Harry Dym ◽

Nicholas Young

Keyword(s):

Unit Circle ◽

Matrix Polynomial ◽

Hermitian Matrix ◽

Krein Space ◽

Gram Matrix ◽

Natural Basis ◽

Test Matrix ◽

Matrix Polynomials ◽

Rational Vector ◽

Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

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