Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Asymptotic behavior of the self-defocusing nonlinear Schrödinger equation for piecewise constant initial conditions

Abstract: In this paper we use a transfer matrix method to calculate the asymptotic behavior of the nonlinear Schrodinger (NLS) equation in a self-defocusing medium for piecewise constant initial conditions. Treating initial conditions that consist of m repeated regions, we show that the eigenvalues associated with this problem appear in bands, and, as m tends to infinity, we obtain the eigenvalue density of states for these bands. Comparing results from the transfer matrix approach to the results for a Bloch function method, we show that the edges of a region with periodic initial conditions result in a finite number of additional eigenvalues that appear outside the bands.

On the Danilewski Method

Abstract: It is a well known fact; that the Danilewski method for finding the characteristic polynomial of a matrix tends to be numerically unstable. (See, for example [3, 4].) Moreover, the roots of a polynomial can vary greatly when a small change is made in the polynomial coefficients. (See, for example, [8].) Hence it has been argued that the Danilewski method should not be used for obtaining matrix : eigenvalues. In this paper we describe a computational procedure for the Danilewski method which employs a \"search for pivots.\" It yields the characteristic polyi: } nomiM much more accurately than the procedures reported by previous investigators. We shall then present a count of mathematical operations used in applying the method with multiple precision arithmetic. These operation counts emphasize the well-known fact that the method is quite fast. We then conclude that the } method should not be regarded as an uneconomical process for obtaining matrix eigenvalues.

Curves on S n -1 that lead to eigenvalues or their means of a matrix

Abstract: This paper discusses two dynamical systems on the unit sphere $S^{n - 1} $ in $\mathbb{R}^n $ space, each defined in terms of a real square matrix M. The solutions of these systems are found to converge to points which provide essential information about eigenvalues of the matrix M. It is shown, in particular, how the dynamics of the second flow is analogous to that of the Rayleigh quotient iterations.

Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable

Abstract: In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.

An inequality for linear transformations with eigenvalues

Abstract: The purpose of this announcement is to state theorems concerning bounded linear transformations on Hilbert space which are far more general than the recent theorems of H. D. Block and W. H. J. Fuchs [2]. Our theorems are more general even in the case that the transformation is a matrix, as in [2]. The basic idea involved in these theorems was first communicated to the author by Professor H. F. Bohnenblust in 1957, and has been applied meanwhile to various perturbation problems for ordinary and partial differential equations, [ l ; 4 ] . The theorems in essentially their present form were enunciated by the author in an unpublished manuscript sent to Professor Bohnenblust in July, 1960.