Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

Upper (lower) bounds of the eigenvalues,spread and the open problems for the real symmetric interval matrices

This work is concerned with exploring the upper bounds and lower bounds of the eigenvalues of real symmetric matrices of order n whose entries are in a given interval. It gives the maximum and minimum of the eigenvalues and the upper bounds of spread of real symmetric interval matrices in all cases. It also gives the answers of the open problems for the maximum and minimum of the eigenvalues of real symmetric interval matrices. Copyright © 2012 John Wiley & Sons, Ltd.     

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.     

Estimates for the real and imaginary parts of the eigenvalues of matrices and applications

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.     

The spread of matrices and polynomials

The spread of a matrix (or polynomial) is the maximum distance between any two of its eigenvalues (or its zeros). E. Deutsch has recently given upper bounds for the spread of matrices and polynomials. We obtain sharper, simpler upper bounds and observe that they are also upper bounds for the sum of the absolute values of the two largest eigenvalues (or zeros).     

The bounds of the smallest and largest eigenvalues for rank-one modification of the Hermitian eigenvalue problem

The bounds of the smallest and largest eigenvalues for rank-one modification of the Hermitian matrices are studied in this paper. The sharper bounds are obtained. Numerical examples illustrate that our bounds give accurate estimates.     

Relative eigenvalue and singular value perturbations of scaled diagonally dominant matrices

The paper derives improved relative perturbation bounds for the eigenvalues of scaled diagonally dominant Hermitian matrices and new relative perturbation bounds for the singular values of symmetrically scaled diagonally dominant square matrices. The perturbation result for the singular values enlarges the class of well-behaved matrices for accurate computation of the singular values. AMS subject classification (2000)  65F15     

Relationship between genetic algebras and semicommutative matrices

Three kinds of noncommutative Gonshor genetic algebras are defined and characterized in terms of matrices. A necessary condition for an algebra to have one of these properties is the semicommutativity of a set of matrices representing the left (and the right) transformations induced by basis elements. For Gonshor genetic algebras which are interpretable, bounds for the train roots of the algebraare given. In terms of matrices this result yields bounds for the eigenvalues of a set ofcertain stochastic semicommutative matrices.     

On real eigenvalues of real tridiagonal matrices

Lower bounds for the number of different real eigenvalues as well as for the number of real simple eigenvalues of a class of real irreducible tridiagonal matrices are given. Some numerical implications are discussed.     

Bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications

We consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of Toeplitz matrices. The Toeplitz structure is destroyed since two elements on each off-diagonal are perturbed. Based on a careful analysis, we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix. In this way, we also obtain a lower bound for the smallest singular value of certain matrices. Some numerical results indicate that our bounds are extremely good.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.     

On eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

In this paper, we give some structured perturbation bounds for generalized saddle point matrices and Hermitian block tridiagonal matrices. Our bounds improve some existing ones. In particular, the proposed bounds reveal the sensitivity of the eigenvalues with respect to perturbations of different blocks. Numerical examples confirm the theoretical results.     

A comparison of some bounds for the nontrivial eigenvalues of stochastic matrices

Summary Some recently published bounds for the nontrivial eigenvalues of stochastic matrices [1, 2, 4, 5] are compared. It is shown that the Deutsch bound [1] is the best of these bounds and is only slightly improved by a bound given in [3].     

Some New Bounds for Singular Values and Eigenvalues of Matrix Products

For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB.     

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.     

Error analysis of floating-point computation

This paper consists of two main sections. In the first the bounds are derived for the rounding errors made in the fundamental floating-point arithmetic operations. In the second, these results are applied in the analysis of a number of computing techniques for the calculation of the eigenvalues of matrices. In each case thecomputed solution is expressed as the exact solution of a perturbed version of the original matrix and bounds are found for the perturbations. For one of the techniques, an a priori bound is derived for the errors in the eigenvalues themselves.Communication from the National Physical Laboratory.     

Extremal graph characterization from the bounds of the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained.     

Homotopy method for the eigenvalues of symmetric tridiagonal matrices

We will present the homotopy method for finding eigenvalues of symmetric, tridiagonal matrices. This method finds eigenvalues separately, which can be a large advantage on systems with parallel processors. We will introduce the method and establish some bounds that justify the use of Newton’s method in constructing the homotopy curves.