Formula omitted]-splittings of matrices

Different convergence and comparison theorems for proper regular splittings and proper weak regular splittings are discussed. The notion of double splitting is also extended to rectangular matrices. Finally, convergence and comparison theorems using this notion are presented.

For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10

Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.

The aim of this paper is to analyze the asymptotic behavior of the eigenvalues and eigenvectors of particular sequences of products involving two square real matrices $A$ and $B$, namely of the form $B^kA$, as $k\rightarrow \infty$. This analysis represents a detailed deepening of a particular case within a general theory on finite families $\mathcal{F} = \{ A_1, \ldots, A_m \}$ of real square matrices already available in the literature. The Bachmann-Landau symbols and related results are largely used and are presented in a systematic way in the final Appendix.

Inequalities concerning real square matrices A with positive definite symmetric component A+A ∗are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite A+A ∗

In this paper the solution of an inverse singular value problem is considered. First the decomposition of a real square matrix A = UΣV is introduced, where U and V are real square matrices orthogonal with respect to a particular inner product defined through a real diagonal matrix G of order n having all the elements equal to ±1, and Σ is a real diagonal matrix with nonnegative elements, called G-singular values. When G is the identity matrix this decomposition is the usual SVD and Σ is the diagonal matrix of singular values. Given a set σ1, ..., σn of n real positive numbers we consider the problem to find a real matrix A having them as G-singular values. Neglecting theoretical aspects of the problem, we discuss only an algorithmic issue, trying to apply a Newton type algorithm already considered for the usual inverse singular value problem.

The article presents the methodological and technological aspects of the theoretical analysis of the first group of motion equations (I. Newton’s dynamic equations), which are the core of the inertial navigation theory and systems. Conceptually, the notion of "analysis" is replaced by another methodological concept "interpretation" which "carries something significantly more important—the understanding that is necessary for producing new ideas" (Academician N. N. Moiseev). The purpose of the work is to verify and develop existing model concepts of motion based on their strict compliance with the axiomatics of Newtonian theory. By referring to the well-known matrix analysis procedure of symmetrization and alternation of a square matrix an expansion of the operator (3Ѕ3 dimension) of the total derivative of the differential motion equation is performed. The efficiency and relevance of the procedure is illustrated by an example of a partial solution of a two-point boundary value problem. The relevance of the decomposition of real square matrices of other dimensions for estimating their characteristic numbers is noted. The forms of the motion equations in various coordinate systems are presented. The general incorrectness of the Newtonian theory interpretation in a model of space built on a system of geodetic coordinates is shown due to the absence of attribute of the corresponding motion equations. At the same time, covariance takes place in a special identified case of motion.

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.

A note on generalizations of strict diagonal dominance for real matrices

We consider E(x)x(k + 1) = F(k)x(k),k ∈ Z + 0, where E(k) and F(k) are real square matrices of order n, not necessarily invertible. Assuming the regularity of the matrix pencils λE(x) - F(k)x(k), k ∈ Z + 0 and the existence of a nonzero common eigenvector of the family of n x n real matrices {E(k) = [α k E(k) - F(k)] -1 E(k), α k ∈ R, k ∈ Z + 0}, we will obtain a solution to the above descriptor system. We also analyse a particular case related with a positive equation - the closed dynamic Leontief model.

Symbolic n-plithogenic sets are considered to be modern concepts that carry within their framework both an algebraic and logical structure. The concept of symbolic n-plithogenic algebraic rings is considered to be a novel generalization of classical algebraic rings with many symmetric properties. These structures can be written as linear combinations of many symmetric elements taken from other classical algebraic structures, where the square symbolic k-plithogenic real matrices are square matrices with real symbolic k-plithogenic entries. In this research, we will find easy-to-use algorithms for calculating the determinant of a symbolic 3-plithogenic/4-plithogenic matrix, and for finding its inverse based on its classical components, and even for diagonalizing matrices of these types. On the other hand, we will present a new algorithm for calculating the eigenvalues and eigenvectors associated with matrices of these types. Also, the exponent of symbolic 3-plithogenic and 4-plithogenic real matrices will be presented, with many examples to clarify the novelty of this work.

We consider the problem to determine the optimal rotations which minimize urn:x-wiley:00442267:media:zamm201800120:zamm201800120-math-0002for a given diagonal matrix with positive entries . The objective function W is the reduced form of the Cosserat shear‐stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy , determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block‐diagonal matrix with blocks of size at most two.

We derive characterizations for the Schur stability and the stability of all convex combinations of k k⩾2, given real square matrices. Moreover, we characterize these properties for the set r(A, B) (c(A,B), resp.) of square matrices whose rows (columns, resp.) are independent convex combinations of the rows (columns, resp.) of two real matrices A and B. Our results can be viewed as contributions to the problem of robustness of matrix properties. This paper continues our paper [4].

Let $K(\alpha ,\beta )$ ($\alpha $, $\beta $ real) be a family of real square matrices. Several computational problems are equivalent to the calculation of a pair $(\alpha ^0 ,\beta ^0 )$ of parameter values for which $K(\alpha ^0 ,\beta ^0 )$ has rank deficiency 2. Among these problems are the computation of a conjugate pair of complex eigenvalues of a given real matrix, the computation of a Hopf bifurcation point on a branch of stationary solutions to a parametrized differential equation, and the computation of a Takens–Bogdanov point in the two-dimensional solution manifold of a set of nonlinear equations. Developing ideas of Griewank and Reddien, the authors define scalar functions $g_1 (\alpha ,\beta ),g_2 (\alpha ,\beta )$ that vanish in $(\alpha ^0 ,\beta ^0 )$. A nondegeneracy condition NDC, which expresses the fact that $(\alpha ^0 ,\beta ^0 )$ is in a natural sense an isolated point in $(\alpha ,\beta )$-space, is introduced. It is proved that under certain conditions on the family $K(\alpha ,\beta )$ the Jacobian of $g_1 $, $g_2 $ with respect to $\alpha $, $\beta $ is nonsingular if and only if NDC holds. A Newton method to compute $(\alpha ^0 ,\beta ^0 )$ is then described. The problems mentioned above and some related ones are analysed in detail. The derived algorithms for the dynamical systems problems have the following features: (1) they are simple and natural, being based on linear algebra concepts only; (2) they treat the two parameters in a symmetric way; (3) they do not lead to formally large systems; and (4) NDC is expressed in terms independent of the particular problem. Numerical results are given which illustrate the quadratic convergence of the Newton algorithm during the computation of a Hopf bifurcation point arising in a model of a tubular reactor.

This paper introduces a matrix analog of the Bessel processes, taking values in the closed set $E$ of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map $x\mapsto x^\top x$. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function $w(x)=|\det x|^\alpha$ is a weight function in the Muckenhoupt $A_p$ class for $-1<\alpha\le 0$ ($p=1$) and $-1<\alpha<p-1$ ($p>1$). The set of matrices of co-rank at least two has zero capacity with respect to the measure $m(dx)=|\det x|^\alpha dx$ if $\alpha>-1$, and if $\alpha\ge 1$ this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) $E$ with Neumann boundary conditions. The highly non-convex, non-Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods.