Related papers: Perturbation theory for Hermitian matrix-functions…

We provide first order perturbation formulas for the matrix square root (in the positive semi-definite case) and the matrix modulus (in the general case). The results are new for singular matrices, and extend previously known Fr\'{e}chet…

We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace…

Let A and E be Hermitian self-adjoint matrices, where A is fixed and E a small perturbation. We study how the eigenvalues and eigenvectors of A+E depend on E, with the aim of obtaining first order formulas (and when possible also second…

We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko on representations for the remainders of the first and second order Taylor-type approximations of operator…

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent…

Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…

In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an…

This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le…

Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their…

The non-Hermitian but $\mathcal{PT}$-symmetric quantum field theories are known to have a pseudo-Hermitian interpretation. However the corresponding intertwining operator happens to be nonlocal that raises the question to what extent this…

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial…

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector…

We extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions. We show that if $f$ is the primitive of a function that is operator monotone on an interval…

This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic…

We give a Herglotz-type representation of an arbitrary generalized spectral measure. As an application, a new proof of the classical Naimark's dilation theorem is given. The same approach is used to describe the spectrum of all unitary…

This work is concerned with variational analysis of so-called spectral functions and spectral sets of matrices that only depend on eigenvalues of the matrix. Based on our previous work [H. T. B\`ui, M. N. B\`ui, and C. Clason, Convex…

The behavior of correlation functions is studied in a class of matrix models characterized by a measure $\exp(-S)$ containing a potential term and an external source term: $S=N\tr(V(M)-MA)$. In the large $N$ limit, the short-distance…

The aim of this article is to present a brief overview of spectral perturbation theory for matrices, bounded linear operators and holomorphic operator-valued functions. We focus on bounds for perturbed eigenvalues, eigenvectors and…

We consider general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$. A distribution matrix-valued function $\Si(s), \; s\in\bR,$ is called a…

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the…