Explicit polar decomposition of complex matrices

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A * and A † commute, Linear and Multilinear Algebra 14 (1984) 241-256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.

arXiv: Rings and Algebras, 2020

This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with commuting factors: First, a proof for a generalisation of the link between polar decomposition with commuting factors and normal matrices is given. Then, some properties of such decompositions are studied and it is shown that the commutativity of the factors only depends on the selfadjoint part. Eventually, polar decompositions with commuting factors are studied under similarity transformations that do not alter the structure of the space. For this purpose, normal forms are decomposed and analysed.

Advances in Difference Equations, 2016

We present several numerical schemes for computing the unitary polar factor of rectangular complex matrices. Error analysis shows high orders of convergence. Many experiments in terms of number of iterations and elapsed times are reported to show the efficiency of the new methods in contrast to the existing ones.

Sarajevo Journal of Mathematics, 2015

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A * and A † commute, Linear and Multilinear Algebra 14 (1984) 241-256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.

Applied Mathematics Letters, 2012

We present an idea for computing complex square roots of matrices using only real arithmetic.

Linear Algebra and its Applications, 1993

We study properties of coninvolutory matrices (EE = I), and derive a canonical form under similarity as well as a canonical form under unitary consimilarity for them. We show that any complex matrix has a coninvolutory dilation, and we characterize the minimum size of a coninvolutory dilation of a square matrix. We characterize the m-by-n complex matrices A that can be factored as A = RE with R real and E coninvolutory, and we discuss the uniqueness of this factorization when A is square and nonsingular. ROGER A. HORN AND DENNIS I. MERINO We denote the set of m-by-n complex matrices by M,,,, and write M, = M The set of m-by-n matrices with real entries is denoted by M,,,@$&d we write M,(IW) = M, ,,([w). A matrix E E M, is said to be coninvolutoy if EE = I, that is, E is'nonsingular and E = E-'. We denote the set of coninvolutory matrices in M, by gn. For A, B E M,, A is similar to B if there exists a nonsingular matrix S E M, such that A = SBS-l, and we write A N B; A and B are said to be consimilar if there exists a nonsingular S E M, such that A = SBS-l. Given a scalar A E C, the n-by-n upper triangular Jordan block corresponding to A is denoted by J,,(h). Topological and metric properties of subsets of M,,,. (compact, bounded, etc.) are always with respect to the topology generated by some norm on M m, n.

Linear Algebra and its Applications, 1996

and &ok-&n Song ABSTRACT A complex square matrix A is called conoertiblc if there is a matrix B obtained h\ A from affixing k signs to entries of A such that per A = det B. In this note it is proved that a complex matrix all of whose entrices are taken from a fixed sector of' angle r/n is convertible if and only if its support is.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

The Cartan decomposition in a semisimple Lie group is a generalization of the polar decomposition of matrices. In this paper we consider an even more general setting in which one obtains an analogous decomposition. In the semisimple case, this decomposition was worked out in a seminal paper of G. I. Ol'shanski 13]. In this paper we give general necessary and su cient conditions for this decomposition to exist in arbitrary real nite dimensional Lie algebras and discuss various contexts and examples where this decomposition obtains, particularly examples related to contraction semigroups.

Mathematica Slovaca, 2019

We introduce a nonsymmetric matrix defined by q-integers. Explicit formulæ are derived for its LU-decomposition, the inverse matrices L−1 and U−1 and its inverse. Nonsymmetric variants of the Filbert and Lilbert matrices come out as consequences of our results for special choices of q and parameters. The approach consists of guessing the relevant quantities and proving them later by traditional means.

Foundations of Computational Mathematics, 2001

The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group.

Linear Algebra and its Applications, 1999

We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results. (R.A. Horn), [email protected] (D.I. Merino) 0024-3795/99/$ -see front matter 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5 ( 9 9 ) 0 0 1 9 9 -8 412 R. A. Horn, D.I. Merino / Linear Algebra and its Applications 302-303 (1999) [411][412][413][414][415][416][417][418][419][420][421] λ; see [2, Chapter 3] and [3, Chapter 6] for basic facts about the Jordan Canonical Form and how it behaves when acted on by a primary matrix function such as the inverse, exponential, log, or square.

Journal of mathematical analysis and …, 2006

Let C(p)C(p) denote the Frobenius companion matrix of the monic polynomial p with complex coefficients. The singular value decomposition and the QR-decomposition of C(p)C(p) are computed.

Linear Algebra and its Applications

Let S∈Mn(R) be such that S2=I or S2=-I. For A∈Mn(C), define ϕS(A)=S-1ATS. We study ϕS-orthogonal matrices (those A∈Mn(C) that satisfy ϕS(A)=A-1). Let F=R or F=C. We show that every ϕS-orthogonal A∈Mn(F) has a polar decomposition A=PU with P,U∈Mn(F),P is positive definite, U is unitary, and both factors are ϕS-orthogonal. We show that if A is ϕS-orthogonal and normal, and if −1 is not an eigenvalue of A, then there exists a normal ϕS-skew symmetric N (that is, ϕS(N)=-N) such that A=eiN. We also take a look at the particular cases S=Hk≡0Ik-Ik0 and S=Lk≡Ik⊕-In-k.

2020

Through this research the following research objectives should be met: * Present spectral factorization of invertible non-scalar matrices ([24] and ) in order to place the current investigation concerning factorization in a broader context. * Present a unified and coherent treatment of the factorization of singular matrices as contained in the works by Wu [29], Laffey [17] and Sourour [26]. * Investigate applications of spectral factorization to invertible matrices i.e. unipotent, positive-definite, commutator, involutory and Hermitian factorization as found in [24], [17], [18] and [26]. * Investigate the conditions under which an invertible matrix can be expressed as a product of two involutions as found in [13]. * Investigate applications of spectral factorization on singular matrices i.e. positive-semidefinite factorization as found in [26]. * The UNISA library catalogue * E-journals * MathSciNet * arXiv () * Wikipedia * JSTOR * Cambridge Journal Online * ScienceDirect