On zero-pattern invariant properties of structured matrices

It is interesting that inverse M-matrices are zero-pattern (power) invariant. The main contribution of the present work is that we characterize some structured matrices that are zero-pattern (power) invariant. Consequently, we provide necessary and sufficient conditions for these structured matrices to be inverse M-matrices. In particular, to check if a given circulant or symmetric Toeplitz matrix is an inverse M-matrix, we only need to consider its pattern structure and verify that one of its principal submatrices is an inverse M-matrix.     

On the eigenvalues of some tridiagonal matrices

A solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by William Trench. The method presented can be generalizable to other problems.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

The eigenvalues of Hermite and rational spectral differentiation matrices

Summary We derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains. In particular, we consider Galerkin and collocation methods based on Hermite functions as well as rational functions (a Fourier series combined with a cotangent mapping). We show that (i) first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues, (ii) for the Hermite method the eigenvalues are determined by the roots of the Hermite polynomials and for the rational method they are determined by the Laguerre polynomials, and (iii) the Hermite method has attractive stability properties in the sense of small condition numbers and spectral radii.     

Tensor properties of multilevel Toeplitz and related matrices

A general proposal is presented for fast algorithms for multilevel structured matrices. It is based on investigation of their tensor properties and develops the idea recently introduced by Kamm and Nagy in the block Toeplitz case. We show that tensor properties of multilevel Toeplitz matrices are related to separation of variables in the corresponding symbol, present analytical tools to study the latter, expose truncation algorithms preserving the structure, and report on some numerical results confirming advantages of the proposal.     

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

On the spectra of sums of orthogonal projections with applications to parallel computing

Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product.This work was supported in part by the Norwegian Research Council for Science and the Humanities under grant D.01.08.054 and by The Royal Norwegian Council for Scientific and Industrial Research under grant IT2.28.28484; also supported in part by the Air Force Office of Scientific Research under grant AFOSR-86-0126 and by the National Science Foundation under grant DMS-8704169.     

Total nonpositivity of nonsingular matrices

In this paper, nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns. These characterizations extend an existing characterization that uses some restrictive hypotheses.     

Eigenvalues of symmetrizable matrices

New perturbation theorems for matrices similar to Hermitian matrices are proved for a class of unitarily invariant norms calledQ-norms. These theorems improve known results in certain circumstances and extend Lu's theorems for the spectral norm, see [Numerical Mathematics: a Journal of Chinese Universities, 16 (1994), pp. 177–185] toQ-norms. This material is based in part upon the third author's work supported from August 1995 to December 1997 by a Householder Fellowship in Scientific Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Office of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp.     

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1≤j≤n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.     

Transition matrices for well-conditioned Markov chains

Let T∈R^n×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where A_j, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ₃ provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ₃ is as small as possible.     

On close eigenvalues of tridiagonal matrices

Summary. A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element and must be a direct sum of two complementary blocks, both of which have the eigenvalue. Yet it is well known that a small spectral gap does not necessarily imply that some is small, as is demonstrated by the Wilkinson matrix. In this note, it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the coupling the two blocks may not be small. In particular, some explanatory bounds are derived and a connection to the Lanczos algorithm is observed. The nonsymmetric problem is also included. Received April 8, 1992 / Revised version received September 21, 1994     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

On the inverses of general tridiagonal matrices

In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to Ostrowski (1952) 23, Shivakumar and Ji (1996) 26, Nabben (1999) [21] and [22] and recently given by Peluso and Politi (2001) 24, Peluso and Popolizio (2008) 25 and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by 2n-2 parameters ( and ) instead of 2n+2 ones as given by El-Mikkawy (2004) 3, El-Mikkawy and Karawia (2006) 4 and Huang and McColl (1997) 10. According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth-death processes together with numerical tests are given.     

Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995     

Note on structured indefinite perturbations to Hermitian matrices

In a recent paper, Overton and Van Dooren have considered structured indefinite perturbations to a given Hermitian matrix. We extend their results to skew-Hermitian, Hamiltonian and skew-Hamiltonian matrices. As an application, we give a formula for computation of the smallest perturbation with a special structure, which makes a given Hamiltonian matrix own a purely imaginary eigenvalue.     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

On a fast and accurate method for computing Fourier transforms

In this paper, a variable order method for the fast and accurate computation of the Fourier transform is presented. The increase in accuracy is achieved by applying corrections to the trapezoidal sum approximations obtained by the FFT method. It is shown that the additional computational work involved is of orderK(2m+2), wherem is a small integer andKn. Analytical expressions for the associated error is also given.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld