Showing papers on "Spectrum of a matrix published in 1982"
TL;DR: The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments as discussed by the authors, which is mainly combinatorial, and it is shown that the sum of eigen values, raised to k -th power, k = 1, 2, 3, 4, 5, 6, m is asymptotically normal.
410 citations
TL;DR: The computed eigenvalues are shown to be the exact eigen values of a matrix M+E where $\Vert E \Vert$ depends on the square root of the machine precision.
199 citations
TL;DR: In this paper, the regular method of the calculation of nonrelativistic hamiltonian eigenfunctions and eigenvalues is developed, and simple and sufficiently precise expressions for eigen values may be found by means of algebraic transformation.
70 citations
TL;DR: In this paper, it was shown that in 3-space variables there are no strictly hyperbolic systems if n s 2(4) is large enough to support singularity propagation.
Abstract: There are many examples of first order n x n systems of partial differential equations in 2 space variables with real coefficients which are strictly hyperbolic; that is, they have simple characteristics. In this note we show that in 3 space variables there are no strictly hyperbolic systems if n s 2(4). Multiple characteristics of course influence the propagation of singularities. For a different context see Appendix 10 of [2]. M denotes the set of all real n x n matrices with real eigenvalues. We call such a matrix nondegenerate if it has n distinct real eigenvalues.
55 citations
TL;DR: In this article, a geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix, which leads to some classical inequalities as well as some new ones.
Abstract: A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if U, V are unitary matrices and K is a skew-Hermitian matrix such that UV~' = exp K, then for every unitary-invariant norm the distance between the eigenvalues of V and those of Vis bounded by \\K\\. This generalises two earlier results which used particular unitary-invariant norms.
28 citations
TL;DR: In this article, the problem of finding the eigenvalues of a certain matrix that is tridiagonal in form was reduced to that of finding a certain eigen function of a positive definite quadratic form.
Abstract: The differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator. Conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial. The problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form. Properties of the eigenvalues of this matrix are investigated, since they are functions of a parameter occurring in the positive definite quadratic form. The asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter.
18 citations
TL;DR: In this article, a modification to the well known bisection algorithm when used to determine the eigenvalues of a real symmetric matrix is presented, where the terms in the Sturm sequence are computed only as long as relevant information on the required eigen values is obtained.
Abstract: A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required.
The technique is also applicable to other computational methods which use the bisection process.
16 citations
TL;DR: In this article, the problem of locating the eigenvalues of A relative to S can be transformed into that for an appropriate block companion matrix relative to R by a rational transformation.
Abstract: Let S be a region which can be mapped into a standard region R of the complex plane (unit circle or half-plane) by a rational transformation. Then the problem of locating the eigenvalues of A relative to S can be transformed into that for an appropriate block companion matrix relative to R .
9 citations
TL;DR: In this paper, the dependence of the eigenvalues of a tridiagonal matrix upon off-diagonal entries was studied and the change in eigenvalue when a cross diagonal product approaches zero or infinity was estimated.
4 citations
TL;DR: In this paper, a symmetric matrix of order n is constructed such that whenever S is nonsingular, A and B do not have an eigenvalue in common, and when S is singular, its nullity is the same as the number of common eigenvalues between A and b.
Abstract: Given two real lower Hessenberg matrices A and B of order n and m (m \leq n) , respectively, a symmetric matrix of order n is constructed such that whenever S is nonsingular, A and B do not have an eigenvalue in common. When S is singular, its nullity, is the same as the number of common eigenvalues between A and B . A well-known classical result on the relative primeness of two polynomials and the associated Bezoutian matrix is included as a special case.
3 citations
TL;DR: In this article, the authors propose an effective (but non-rigorous) formula for lower bounds to eigenvalues of a Hermitian operator, which obviates the need for a priori knowledge of the spectrum.
Abstract: The authors propose an effective (but non-rigorous) formula for lower bounds to eigenvalues of a Hermitian operator. The procedure is similar to that of Temple, but obviates the need for a priori knowledge of the spectrum. It yields very accurate results for two-electron atoms and ions.
01 Jan 1982
TL;DR: In this article, a simple explicit convergence criterion is given as well as the algorithm and two numerical examples for those eigenvalues of a lambda-matrix, which are the values of lambda that make the matrix singular.
Abstract: The matrix N(lambda) whose elements are functions of a parameter lambda is called the lambda-matrix. Those values of lambda that make the matrix singular are of great interest in many applied fields. An efficient method for those eigenvalues of a lambda-matrix is presented. A simple explicit convergence criterion is given as well as the algorithm and two numerical examples.