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A remarkable mathematical property -- somehow hidden and recently rediscovered -- allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. That opens the possibility to get the wavefunctions from the…
The two-dimensional hydrogen-like atom in a constant magnetic field is considered. It is found that this is actually two separate problems. One for which the magnetic field causes an effective attraction between the nucleus and the electron…
This paper examines the properties of real symmetric square matrices with a constant value for the main diagonal elements and another constant value for all off-diagonal elements. This matrix form is a simple subclass of circulant matrices,…
Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonnegative matrix with maximal eigenvalue one and associated unique positive left and right eigenvectors, the article studies the properties of an…
To locate all eigenvalues of a matrix more precisely, we exclude some sets which do not include any eigenvalue of the matrix from the well-known Brauer set to give two new Brauer-type eigenvalue inclusion sets. And it is also shown that the…
Matrix elements between shifted correlated Gaussians of various potentials with several form-factors are calculated analytically. Analytic matrix elements are of importance for the correlated Gaussian method in quantum few-body physics.
We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and…
We analyze the problem of comparing unitary transformations. The task is to decide, with minimal resources and maximal reliability, whether two given unitary transformations are identical or different. It is possible to make such…
For the coefficients of the amplitude a set of simultaneous equations is derived in momentum space. By the auxiliary conditions they are equivalent to nonrelativistic equations and suitable for the investigation of two-nucleon system.
The purpose of this note is twofold: firstly to improve the known results on variation of extreme eigenvalues of birth and death matrices and random walk matrices; and secondly to progress towards the solution of a thirty years old open…
We present a matrix version of a known method of constructing common eigenvectors of two diagonalizable commuting matrices, thus enabling their simultaneous diagonalization. The matrices may have simple eigenvalues of multiplicity greater…
We describe a method for the calculation of accurate energy eigenvalues and expectation values of observables of separable quantum-mechanical models. We discuss the application of the approach to one-dimensional anharmonic oscillators with…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…
We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum…
For states of quantum systems of $N$ particles with harmonic interactions we prove that each reduced density matrix $\rho$ obeys a duality condition. This condition implies duality relations for the eigenvalues $\lambda_k$ of $\rho$ and…
Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P=\{C_{1},\dots,C_{\ell}\}$ of index set $\{1,\dots,n\}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k…
Some monotone increasing sequences of the lower bounds for the minimum eigenvalue of $M$-matrices are given. It is proved that these sequences are convergent and improve some existing results. Numerical examples show that these sequences…
We present a kind of construction for a class of special matrices with at most two different eigenvalues, in terms of some interesting multiplicators which are very useful in calculating eigenvalue polynomials of these matrices. This class…
We investigate self-adjoint matrices $A\in\mathbb{R}^{n,n}$ with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group…
We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the…