Related papers: Differentiating Generalized Eigenvalues and Eigenv…
In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal…
We generalize $\epsilon$-pseudospectra and the associated computational algorithms to the generalized eigenvalue problem. Rank one perturbations are used to determine the $\epsilon$-pseudospectra.
This paper presents the forward and backward derivatives of partial eigendecomposition, i.e. where it only obtains some of the eigenpairs, of a real symmetric matrix for degenerate cases. The numerical calculation of forward and backward…
We present a method to construct a basis of singular and non-singular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with…
We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite…
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that…
Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…
We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian…
Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue…
Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…
A generalized eigenvector of a hypermatrix, called the universal (U-) eigenvector, is proposed, which extended the notion of diagonal (D-) eigenvectors in the literature. Using the semi-tensor product, the homogeneous U-eigenequation can be…
Starting from a mistake done by a student, we discover an unexpected method of finding both eigenvectors for a $2\times2$ matrix with distinct eigenvalues in a single computation. We discuss a connection with the Cayley-Hamilton theorem,…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
In this paper, we derive new model formulations for computing generalized singular values of a Grassman matrix pair. These new formulations make use of truncated filter matrices to locate the $i$-th generalized singular value of a Grassman…
This is an introductory note concerning the distribution vectors in a unitary representation of a Lie group. We discuss the definition of matrix coefficients associated with a pair of distributions and how one can compute them. Most of the…
We report a new analytical method for solution of a wide class of second-order differential equations with eigenvalues replaced by arbitrary functions. Such classes of problems occur frequently in Quantum Mechanics and Optics. This approach…
In this paper, we study spectral properties of generalized weighted Hilbert matrices. In particular, we establish results on the spectral norm, determinant, as well as various relations between the eigenvalues and eigenvectors of such…
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…