Stability properties of superoptimal preconditioner from numerical range

A matrix is said to be stable if the real parts of all the eigenvalues are negative. In this paper, for any matrix A_n, we give some sufficient and necessary conditions for the stability of superoptimal preconditioner E_U(A_n) proposed by Tyrtyshnikov (SIAM J. Matrix Anal. Appl. 1992; 13 :459–473). Copyright © 2005 John Wiley & Sons, Ltd.     

Ein Verfahren zur Stabilitätsfrage bei Matrizen-Eigenwertproblemen

Summary Given the eigenvalue problem (A–E) x=0 for real or complex matricesA the number of eigenvalues with positive real parts is determined without evaluating the caracteristical polynomial. A proceeding is developed here to transform the given matrixA into a reduced form by applying a finite series of elementary transformations upon the matrix. The elements of the reduced matrix allow immediately to solve the problem.     

On a matrix associated to a matrix pair

For a c–stable matrix pair (E,A), i.e., all finite eigenvalues of (E,A) have negative real part, an associated c–stable matrix is provided. It has all finite eigenvalues of (E,A) as eigenvalues and an additional stable eigenvalue −α, where α may be chosen arbitrarily, which corresponds to the eigenvalue ∞ of (E,A). In the case of positive systems, this associated c–stable matrix is, in addition, shown to be a −M-matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A^?1. We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. Copyright © 2009 John Wiley & Sons, Ltd.     

A binary powering Schur algorithm for computing primary matrix roots

An algorithm for computing primary roots of a nonsingular matrix A is presented. In particular, it computes the principal root of a real matrix having no nonpositive real eigenvalues, using real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith algorithm.     

The minimal polynomial of a matrix ∗

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A (λ) of A over F. By using q_A (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications

A square matrix A of order n is said to be tripotent if A ³?=?A. In this note, we give a nine-term disjoint idempotent decomposition for the linear combination of two commutative tripotent matrices and their products. Using the decomposition, we derive some closed-form formulae for the eigenvalues, determinant, rank, trace, power, inverse and group inverse of the linear combinations. In particular, we show that the linear combinations of two commutative tripotent elements and their products can produce 3⁹?=?19,683 tripotent elements.     

The least real part eigenvalue and principal submatrices

We ask for what A is the minimum of the real parts of the eigenvalues bounded above by the corresponding number maximized over principal submatrices of size one less.     

Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T?', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T?' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.     

Exact solutions of the modified Korteweg-de Vries equation

We use the inverse scattering method to obtain a formula for certain exact solutions of the modified Korteweg-de Vries (mKdV) equation. Using matrix exponentials, we write the kernel of the relevant Marchenko integral equation as W( x + y;t ) = Ce ^{- ( x + y )A} e^{8A3 t} B\Omega \left( {x + y;t} \right) = Ce^{ - \left( {x + y} \right)A} e^{8A^3 t} BB, where the real matrix triplet (A,B,C) consists of a constant p×p matrix A with eigenvalues having positive real parts, a constant p×1 matrix B, and a constant 1× p matrix C for a positive integer p. Using separation of variables, we explicitly solve the Marchenko integral equation, yielding exact solutions of the mKdV equation. These solutions are constructed in terms of the unique solution P of the Sylvester equation AP + PA = BC or in terms of the unique solutions Q and N of the Lyapunov equations A^°Q + QA = C^°C and AN + NA^° = BB^°, where B^°denotes the conjugate transposed matrix. We consider two interesting examples.     

Remarks on Non-Linear Spectral Perturbation

We are interested in some aspects of the perturbation effects in the spectrum of a real nonnormal matrix A under linear perturbations. We discuss some known results and we use them to justify some recent experimental observations. Moreover, we demonstrate that the qualitative behavior of the eigenvalues of A under linear perturbations may be predicted by inspecting the spectral radius of a related matrix. Then, we show how this information can be used to analyze the quality of the approximation of a projection method and to justify the presence of unexpected approximate eigenvalues.     

Outlier Eigenvalues for Deformed I.I.D. Random Matrices

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/N. Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.     

Stability and convex hulls of matrix powers

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.     

Powers of a matrix and a formula for the moduli of its eigenvalues

The spectral radius of a complex square matrix A is given by ρ(A) = lim sup_{k → ∞} (TrA^k)^1/k. A more general result is proved which gives information about the moduli of all eigenvalues of A.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison between the convergence rates of the Chebyshev method and the related (2,2)‐step methods

An optimal Chebyshev method for solving A x = b , where all the eigenvalues of the real and non‐symmetric matrix A are located in the open right half plane, is dependent on an optimal ellips∂Ω^* such that the spectrum of A is contrained in Ω^*, the closed interior of the ellipse. The relationship between the convergence rates of the Chebyshev method and the closely related (2,2)‐step iterative methods are studied. (2,2)‐step iterative methods are faster than an optimal Chebyshev method under certain conditions. A numerical example illustrates such an improvement of a (2,2)‐step iterative method. Copyright © 2000 John Wiley & Sons, Ltd.     

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25 :362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A ? Ã‖₂ ? ‖A ? Ã‖_F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14 :588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.