Related papers: Singular Values and Eigenvalues of Tensors: A Vari…

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…

We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their…

We prove an abstract instability result for an eigenvalue problem with parameter. We apply this criterion to show the transverse linear instability of solitary waves on various examples from mathematical physics.

Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric…

The largest eigenvalue of random tensors is an important feature of systems involving disorder, equivalent to the ground state energy of glassy systems or to the injective norm of quantum states. For symmetric Gaussian random tensors of…

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum…

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define…

Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as…

In this paper, we consider the operator properties of various phononic eigenvalue problems. We aim to answer some fundamental questions about the eigenvalues and eigenvectors of phononic operators. These include questions about the…

Quantum--mechanical multiple--well oscillators exhibit curious complex eigenvalues that resemble resonances in models with continuum spectra. We discuss a method for the accurate calculation of their real and imaginary parts.

A new property, the strong singular value property, is introduced, developed, and utilized to study the problem of which lists of nonnegative real numbers occur as the singular values of a matrix with a prescribed zero-nonzero pattern.

In this paper, we find bounds for the eigenvalues of matrix polynomials. In particular, we find generalizations of Cauchy's classical Theorem for distribution of eigenvalues of matrix polynomial.

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=\alpha/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we…

The tensor t-product, introduced by Kilmer and Martin [26], is a powerful tool for the analysis of and computation with third-order tensors. This paper introduces eigentubes and eigenslices of third-order tensors under the t-product. The…

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the…

This paper discusses a general method for spectral type theorems using metric spaces instead of vector spaces. Advantages of this approach are that it applies to genuinely non-linear situations and also to random versions. Metric analogs of…

Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a…

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…

Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for…

Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix…