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We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided in the considered algorithm. A modification is introduced to deal properly with the $J$-skew symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed.

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a spectral perturbation theorem for rank-one updated matrices of special structure in [7] and considered two applications. Cheng, Luo, and Li considered the bounds of the smallest and largest eigenvalues for rank-one modification of Hermitian matrices [8]. Eigenvalue bounds for perturbations of Hermitian matrices have been considered by Ipsen and Nadler in [9]. In this paper, we consider the bounds of the eigenvalues for rank-one modification of Hermitian matrices. The ideas of this paper were mainly motivated by one of [9]. We study the following form

SUMMARY A Hermitian matrix X is called a least-squares solution of the inconsistent matrix equation AXA* = B, where B is Hermitian. A* denotes the conjugate transpose of A if it minimizes the F-norm of B − AXA*; it is called a least-rank solution of AXA* = B if it minimizes the rank of B − AXA*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.

Through a Hermitian-type (skew-Hermitian-type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240:199–205), where A is Hermitian (skew-Hermitian), we show how to find a Hermitian (skew-Hermitian) matrix X such that the matrix expressions A − BX ± X*B* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X*B* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ2V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least-squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least-squares approximation over the cone of positive semi-definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26(3):537–556), who obtained a numerical solution to the real-valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix-dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over-parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.

We further generalize the technique for constructing the Hermitian/skew-Hermitian splitting (HSS) iteration method for solving large sparse non-Hermitian positive definite system of linear equations to the normal/skew-Hermitian (NS) splitting obtaining a class of normal/skew-Hermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive-overrelaxation (SOR) acceleration scheme for the NSS iteration, which specifically results in an acceleration scheme for the HSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. A numerical example is used to show that the SOR technique can significantly accelerate the convergence rate of the NSS or the HSS iteration method. Copyright © 2006 John Wiley & Sons, Ltd.

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F-norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above-mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper we consider solving Hermitian Toeplitz systems Tnx=b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2π-periodic function ƒ, i.e. Tn=n[ƒ]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if ƒ is positive then the spectrum of n[1/ƒ]n[ƒ] is clustered around 1. We prove that the trigonometric polynomial q (s⩾2, cf. (2) and (3)) converges to 1/ƒ uniformly as n→∞ under the condition that 1/ƒ is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/ƒ with q, where N

Summary The contour integral-based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non-Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non-Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design-efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

We are interested in the calculation of explicit formulae for the condition numbers of the two factors of the polar decomposition of a full rank real or complex m × n matrix A, where m ≥ n. We use a unified presentation that enables us to compute such condition numbers in the Frobenius norm, in cases where A is a square or a rectangular matrix subjected to real or complex perturbations. We denote by σ1 (respectively σn) the largest (respectively smallest) singular value of A, and by K(A) = σ1/σn the generalized condition number of A. Our main results are that the absolute condition number of the Hermitian polar factor is √2(1 + K(A)2)1/2/(1 + K(A)) and that the absolute condition number of the unitary factor of a rectangular matrix is 1/σn. Copyright © 2000 John Wiley & Sons, Ltd.

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.