Computing Accurate Eigenvalues of a Hermitian Matrix

1989 ◽  
pp. 104-115
Author(s):  
K. V. Fernando ◽  
M. W. Pont
Keyword(s):  
Hermitian Matrix

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

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.

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

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.

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
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.

An Algorithm for Factoring Hermitian Matrix Polynomials Relative to the Imaginary Axis

1996 ◽  
Vol 29 (1) ◽  
pp. 1263-1268
Author(s):  
Chyi Hwang ◽  
Bo-Win Lin ◽  
Tong-Yi Guo
Keyword(s):  
Imaginary Axis ◽  
Hermitian Matrix ◽  
Matrix Polynomials

scholarly journals Sign Characteristics of Regular Hermitian Matrix Pencils under Generic Rank-1 and Rank-2 Perturbations

2015 ◽  
Vol 30 ◽  
pp. 760-794 ◽  
Author(s):  
Leonhard Batzke
Keyword(s):  
Canonical Form ◽  
Hermitian Matrix ◽  
Structure Preserving ◽  
Spectral Behavior ◽  
Generic Rank ◽  
Jordan Structure ◽  
Matrix Pencils ◽  
Rank 2 ◽  
Infinite Blocks

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

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