Cayley digraphs with normal adjacency matrices

In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form for some non-negative integer n, where Q₈ is the quaternion group and is the abelian group of order 2ⁿ and exponent 2.     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Decompositions of some types of quaternionic matrices

The structures of some important types of matrices over the quaternion skew field are discussed, and corresponding decompositions are obtained by virtue of real orthogonal matrices and real anti-symmetric matrices. In particular, the normal forms are found for several classes of quaternionic matrices.     

线性流形上对称正交反对称矩阵反问题的最小二乘解

设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.     

On the eigenvalues of matrices with prescribed upper triangular part

Consider a matrix D=[D_i,j] partitioned in submatrices D_i,j where the submatrices D_i,iare square. This paper studies the possible eigenvalues of D, when the blocks D_i,j, with i≤j, are fixed, and the other blocks vary. The result obtained generalizes twotheorems already known.     

Orthogonal polyanalytic polynomials and normal matrices

The Hermitian Lanczos method for Hermitian matrices has a well-known connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of . This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials. Denoting by the number of elements computed so far, the arising scheme yields a recurrence of length bounded by for polyanalytic polynomials orthogonal on a discrete subset of . Like this slowly growing length of the recurrence, the method preserves, at least partially, the properties of the Hermitian Lanczos method. We employ the algorithm in least squares approximation and bivariate Lagrange interpolation.

     

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

On matrices that admit unitary reduction to band form

It is proved in the paper that any low rank perturbation of a Hermitian matrix is unitarily reducible to band form. Moreover, if a normal matrix is unitarily reducible to band form, then any of its rank one perturbations is unitarily reducible as well.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 871–880, December, 1998.     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

Research problem: on the norms of infinite Cauchy-Toeplitz-plus-Cauchy-Hankel matrices

We give a conjecture and research problem related to ℓ_p operator and spectral norms of the matrix T_n + H_n.     

Controllability completion problems of partial upper triangular matrices

The main result of this paper states sufficient conditions for the existence of a completion A_c of an n × n partial upper triangular matrix A, such that the pair (A_cB) has prescribed controllability indices, being B an n×m matrix. If A is a partial Hessenberg matrix some conditions may be dropped. An algorithm that obtains a completion A_c of A such that pair (A_ce_k) is completely controllable, where e_k is a unit vector, is used to proof the results.     

Block matrices and symmetric perturbations

We prove that if A=[A_ij]∈R^N,N is a block symmetric matrix and y is a solution of a nearby linear system (A+E)y=b, then there exists F=F^T such that y solves a nearby symmetric system (A+F)y=b, if A is symmetric positive definite or the matricial norm μ(A)=(‖A_ij‖₂) is diagonally dominant. Our blockwise analysis extends existing normwise and componentwise results on preserving symmetric perturbations (cf. [J.R. Bunch, J.W. Demmel, Ch. F. Van Loan, The strong stability of algorithms for solving symmetric linear systems, SIAM J.Matrix Anal. Appl. 10 (4) (1989) 494-499; D. Herceg, N. Kreji?, On the strong componentwise stability and H-matrices, Demonstratio Mathematica 30 (2) (1997) 373-378; A. Smoktunowicz, A note on the strong componentwise stability of algorithms for solving symmetric linear systems, Demonstratio Mathematica 28 (2) (1995) 443-448]).     

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.     

Research problem: indefinite inner product normal matrices

Several open research problems are formulated concerning normal matrices with respect to indefinite inner products.     

Research problem: indefinite inner product normal matrices

Several open research problems are formulated concerning normal matrices with respect to indefinite inner products.     

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.     

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).     

On unitary analogs of GCD reciprocal LCM matrices

A divisor d ∈ ?₊ of n ∈ ?₊ is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM.