Showing papers on "Spectrum of a matrix published in 1987"

Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations.

Abstract: In stability analysis of nonlinear systems, the character of the eigenvalues of the Jacobian matrix (i.e., whether the real part is positive, negative, or zero) is needed, while the actual value of the eigenvalue is not required. We present a simple algebraic procedure, based on the Routh-Hurwitz criterion, for determining the character of the eigenvalues without the need for evaluating the eigenvalues explicitly. This procedure is illustrated for a system of nonlinear ordinary differential equations we have studied previously. This procedure is simple enough to be used in computer code, and, more importantly, makes the analysis possible even for those cases where the secular equation cannot be solved.

Variation of discrete spectra

Abstract: A formula [see (1) below] estimating collectively the variation of eigenvalues of a symmetric matrix under a perturbation is extended to the case of discrete eigenvalues of a selfadjoint operator in Hilbert space, under the assumption that the perturbation is compact. For this purpose, the notion of an extended enumeration of discrete eigenvalues is introduced.

Computation of Eigenvalues associated with functional differential equations

Abstract: This paper describes a method for computing the eigenvalues associated with systems of linear retarded functional differential equations (RFDE's). The method finds the eigenvalues directly from a certain characteristic equation which is automatically determined from system matrices. The eigenvalues contained in some bounded region around the origin are approximately computed by a combinatorial algorithm suggested earlier by H. Kuhn [15] for approximations of zeros of ordinary polynomials. The eigenvalues of large modulus, which are distributed in some curvilinear strips, are computed from some asymptotic formulas obtained directly from the parameters of the characteristic equation. To verify that all the eigenvalues have been found, we use a highly reliable procedure proposed by Carpentier and Dos Santos, which evaluates the number of zeros of an analytic function in a given region. Numerical results are presented for several examples and compared with those obtained by a method based on finite-dimensional approximations of delay equations.

On the eigenvalues of interval matrices

Abstract: In this paper we will examine how the spectrum (of eigenvalues) of an interval matrix family M depends on the spectrum of its extreme sets. We present three results: 1) We show that the roots of pairwise convex combinations of the characteristic polynomials of the vertices of M provide a bound for the spectrum of M. 2) We state conditions on M which guarantee that the spectrum of M can be determined from the spectrum of its relative boundary. 3) For the special case of M having a real spectrum, we show that this spectrum is completely determined by the eigenvalues of its vertices.

A formula for the determinant of a sum of matrices

Abstract: We give a formula, involving circular words and symmetric functions of the eigenvalues, for the determinant of a sum of matrices. Theorem of Hamilton-Cayley is deduced from this formula.

An eigenvalue recursion for Toeplitz matrices

Abstract: In this correspondence, a recursive algorithm for finding eigenvalues of a real symmetric Toeplitz matrix from the eigenvalues of nested Toeplitz submatrices is presented. Given the eigenvalues of two nested Toeplitz submatrices, the eigenvalues of the next larger Toeplitz matrix can be found simply by solving a Vandermonde set of equations and rooting two polynomials.

State-space discrete systems: 2-D eigenvalues

Abstract: A correction is given to a proof of the fact, presented in a paper by Roesser [1], that every partitioned matrix A not only satisfies the two-dimensional characteristic equation, but it must also satisfy an additional set of equations. A new definition of 2-D eigenvalues is proposed.

An integral eigenvalue approach to three-dimensional field problems--A low-frequency variation of SEM

Abstract: The three-dimensional (3-D) eddy-current transient field problem is formulated first using the u-v method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null-field integral equations and the appropriate boundary conditions germane to the problem are used to set up an identification matrix which is independent of null-field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. The eigenvalues, which can be equated with temporal response, are found to be intimately linked to the initial forcing function which triggers the transient in question. When this initial forcing function is Fourier decomposed into its respective spatial harmonics, it is possible to associate with each Fourier component a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response so obtained with the particular imposed external field governing the problem at hand. The technique is applied to the FELIX medium cylinder (a conducting cylinder placed in a collapsing external field) and compared to data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure. The technique proposed is applied in the low-frequency regime where the near-field effects must be considered. Application of the technique to a high frequency follows directly if the Coulomb gauge is adopted to represent the vector potential.

The spectra of matrices having sums of principle minors with alternating sign

Abstract: We present an observation on the localization of the spectrum of a matrix having sums of principal minors with alternating sign.

Inequalities for eigenvalues of the polyharmonic operator Δ p

Abstract: In this paper we consider the bounds of the eigenvalues for a class of polyharmonic operators and obtain the bounds for (n+1)th eigenvalue interm of the firstn eigenvalues Those estimates do not depend on the domain in which the problem is considered