A Comparison of Three Variants of the Generalized Davidson Algorithm for the Partial Diagonalization of Large Non-Hermitian Matrices

2010 ◽  
Vol 6 (7) ◽  
pp. 1966-1970 ◽  
Author(s):  
Marco Caricato ◽  
Gary W. Trucks ◽  
Michael J. Frisch
Keyword(s):  
Hermitian Matrices ◽  
Davidson Algorithm

An algorithm for eigenvectors of non-hermitian matrices

1969 ◽  
Author(s):  
Albert Maurice Erisman
Keyword(s):  
Hermitian Matrices

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde
Keyword(s):  
Finite Volume ◽  
Chaotic System ◽  
Path Integral ◽  
Hermitian Matrices ◽  
The Matrix ◽  
Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

2005 ◽  
Vol 129 (2) ◽  
pp. 3772-3786
Author(s):  
L. Yu. Kolotilina
Keyword(s):  
Hermitian Matrices ◽  
Extreme Eigenvalues

Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies

2014 ◽  
Vol 141 (12) ◽  
pp. 124104 ◽  
Author(s):  
Yang Yang ◽  
Degao Peng ◽  
Jianfeng Lu ◽  
Weitao Yang
Keyword(s):  
Random Phase Approximation ◽  
Random Phase ◽  
Phase Approximation ◽  
Excitation Energies ◽  
Davidson Algorithm

scholarly journals A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Necessary And Sufficient ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

scholarly journals Graphs and Hermitian matrices: eigenvalue interlacing

2004 ◽  
Vol 289 (1-3) ◽  
pp. 119-127 ◽  
Author(s):  
Béla Bollobás ◽  
Vladimir Nikiforov
Keyword(s):  
Hermitian Matrices

scholarly journals Spherical Functions Approach to Sums of Random Hermitian Matrices

2017 ◽  
Vol 2019 (4) ◽  
pp. 1005-1029 ◽  
Author(s):  
Arno B J Kuijlaars ◽  
Pablo Román
Keyword(s):  
Spherical Functions ◽  
Hermitian Matrices

Spectral Distribution of Sequences of Perturbed Hermitian Matrices

2017 ◽  
pp. 57-64
Author(s):  
Carlo Garoni ◽  
Stefano Serra-Capizzano
Keyword(s):  
Spectral Distribution ◽  
Hermitian Matrices

scholarly journals The validity of the Marcus-de Oliveira conjecture for essentially Hermitian matrices

1994 ◽  
Vol 197-198 ◽  
pp. 411-427 ◽  
Author(s):  
Natália Bebiano ◽  
Alexander Kovačec ◽  
João da Providência
Keyword(s):  
Hermitian Matrices
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