Showing papers on "Spectrum of a matrix published in 1970"

Eigenvalues and Eigenfunctions of the Spheroidal Wave Equation

Abstract: A technique is presented for the calculation of the oblate and prolate spheroidal wave equation eigenvalues and eigenfunctions. The eigenvalue problem is cast in matrix form and a tridiagonal, symmetric matrix is obtained. This formulation permits the immediate calculation of the eigenvalues to the desired accuracy by means of the bisection method. The eigenfunction expansion coefficients are then obtained by a recursion method. This technique is quite simple to program, and the computation speed is rapid enough to allow its use as a function subroutine where values not previously tabulated or large numbers of values are required.

Graphs with a small number of distinct eigenvalues

Abstract: Given a loopless graph G with n v e r t i c e s and no mult iple edges, the adjacency matr ix of G, A(G), is a square 0-1 matrix of order n whose rows and columns correspond t o the v e r t i c e s of G and f o r which A 1 i f and only i f t h e i t h and j t h v e r t i c e s a r e adjacent. ijPeigenvalues of t h i s matrix a r e the eigenvalues of the graph. The re la t ionships between a graph and i t s eigenvalues a r e cur ren t ly under inves t iga t ion , and what at, f i r s t glance might appear t o be a somewhat a r t i f i c i a l l y constructed a rea of study has, i n f a c t , proven t o be q u i t e in te res t ing . adjacency matrix have turned out t o be in t imate ly r e l a t e d t o the fami l ia r topological proper t ies of a graph. two quest ions a r e invest igated: (1) Given a s e t of eigenvalues, i s there a graph whose spectrum cons is t s of prec ise ly t h a t s e t , and (2) t h a t spectrum or a r e there others? be made concerning the second question: i f two graphs a r e isomorphic, then t h e i r adjacency matr ices a r e s imilar and they have i d e n t i c a l spectra. Thus throughout t h i s paper two d i f f e r e n t graphs a r e ones t h a t a r e non-isomorphic. The

Perturbation bounds for means of eigenvalues and invariant subspaces

Abstract: When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix. For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix. A numerical example is given.

The one- and two-dimensional quartic oscillators

Abstract: Accurate eigenvalues and eigenvectors of the one- and two- dimensional quartic oscillators have been determined using the linear variation method in which the Hamiltonian matrix was set up in the representation of the corresponding harmonic oscillator. The Hamiltonian matrix was factorized and the submatrices were diagonalized using two different methods: (i) Matrices of order 100 were diagonalized giving both eigenvalues and eigenvectors. (ii) Matrices of order 800 were diagonalized giving eigenvalues only. The former of these procedures yields eigenvalues accurate to nine figures. Mixed harmonic-quartic potentials have been investigated for the two-dimensional case.

Brackets to the eigenvalues of the Schrödinger equation, part 1. Tridiagonal matrices

Abstract: The problem of upper and lower bounds to the first few eigenvalues of a very large or infinite tridiagonal matrix H is studied. Those eigenvalues of a comparison-matrix Mn which are lower than a characteristic limit, together with the corresponding eigenvalues of the variational matrix Hn are shown to bracket exact eigenvalues of H. Mn differs from Hn only in the last off-diagonal element and is easily obtained from H. Sufficient conditions for lower bounds are based on a low estimate of the characteristic limit. For increasing dimensions n, the lower bounds approach the exact eigenvalues from below. As a numerical illustration, brackets to the known eigenvalues of the harmonic oscillator with a linear perturbation are calculated.

Criteria for matrix regularity and spectrum localization

Abstract: Criteria for a matrix to be non-singular are outlined below, expressed both in terms of the moduli of the matrix elements and minors, and in terms of the boundary norms of its blocks. The criteria are used to find the regions in which the matrix spectrum is localized.

Elimination of zero eigenvalues from the A matrix for linear RLC networks

Abstract: This correspondence deals with the elimination of zero eigenvalues of the A matrix for linear time-invariant RLC networks and the consequent reduction of order of the state equations. The state variables used are a subset of the capacitor voltages and inductor currents, chosen such that the order of the state equation is equal to the number of finite nonzero eigenvalues or natural frequencies of the network. Two methods are given for the reduction to minimum order of the commonly used linear state equation.

Triangular decomposition as an aid in determining eigenvalues of large-order banded symmetric matrices

Abstract: Triangular decomposition as aid in determining eigenvalues of large-order banded symmetric matrices