The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX?+?BX ^??=?0, where A, B?∈?? ^m×n are two given matrices, X?∈?? ^n×n is an unknown matrix, and ? denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A?+?λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A?+?λB to the canonical form by strict equivalence.     

Specht property for the 2-graded identities of the Jordan algebra of a bilinear form

Let K be a field of characteristic zero and V a vector space of dimension m>1 with a nondegenerate symmetric bilinear form f:V×V→K. The Jordan algebra B_m = K⊕V of the form f is a ?₂-graded algebra with this decomposition. We prove that the ideal of all the ?₂-graded identities of B_m satisfies the Specht property and we compute the ?₂-graded cocharacter sequence of B_m.     

Anisotropic Modules over Artinian Principal Ideal Rings

Let V be a finite-dimensional vector space over a field k, and let W be a 1-dimensional k-vector space. Let ?,?: V × V → W be a symmetric bilinear form. Then ?,? is called anisotropic if for all nonzero v ∈ V we have ? v, v ? ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss the radical root of a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory.     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

The Constrained Solutions of Two Matrix Equations

We study the symmetric positive semidefinite solution of the matrix equation AX ₁ A ^T + BX ₂ B ^T = C, where A is a given real m×n matrix, B is a given real m×p matrix, and C is a given real m×m matric, with m, n, p positive integers; and the bisymmetric positive semidefinite solution of the matrix equation D ^T XD = C, where D is a given real n×m matrix, C is a given real m×m matrix, with m, n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions. Received December 17, 1999, Revised January 10, 2001, Accepted March 5, 2001     

Divisibility of multilinear mappings with applications to affine differential geometry

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h?:?V?×?V?→?W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C?:?V?×?V?×?V?→?W is divisible by h if there exists a linear form α such that C(v,?v,?v)?=?α(v)h(v,?v) for every v?∈?V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v,?v)?∧?C(v,?v,?v)?=?0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ₁,λ₂,λ₃,σ) from a given realizable list of n complex numbers (c ₁,c ₂,σ), where c ₁ is the Perron eigenvalue, c ₂ is a real number and σ is a list of n???2 complex numbers.     

Backward Perturbation Analysis for the Matrix Equation A~TXA+B~TYB=D

Consider the linear matrix equation A~TXA + B~TYB = D,where A,B are n X n real matrices and D symmetric positive semi-definite matrix.In this paper,the normwise backward perturbation bounds for the solution of the equation are derived by applying the Brouwer fixed-point theorem and the singular value decomposition as well as the property of Kronecker product.The results are illustrated by two simple numerical examples.     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type A _n , B _n , C _n , D _n , F ₄ and G ₂ over \mathbbC\mathbb{C}, the projective variety \mathbbP(V^m)/W\mathbb{P}(V^m)/W is projectively normal with respect to the descent of O(1)^?|W|\mathcal{O}(1)^{\otimes |W|}, where V ^m denote the direct sum of m copies of V.     

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.     

Generalizations of the odd degree theorem and applications

LetV ⊂ ℙℝ ⁿ be an algebraic variety, such that its complexificationV _ℂ ⊂ ℙ ⁿ is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝ ⁿ of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX?XB+C=0 where the matrix C∈?^n×m is of low rank and the spectra of A∈?^n×n and B∈?^m×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X? of rank k=O(log(1/ε)) such that ∥X?X?∥₂?ε∥X∥₂. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solutions of AXB = C for centrosymmetric matrix X under a specified submatrix constraint

We say that X = [x_ij] is centrosymmetric if x_ij = x_{n ? j + 1, n ? i + 1}, 1?i, j?n. In this paper, we present an efficient algorithm for minimizing ∥AXB ? C∥ where ∥·∥ is the Frobenius norm, A∈?^{m × n}, B∈?^{n × s}, C∈?^{m × s} and X∈?^{n × n} is centrosymmetric with a specified central submatrix [x_ij]_{p?i, j?n ? p}. Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable in any case, and we give results of numerical experiments that support this claim. Copyright © 2011 John Wiley & Sons, Ltd.     

On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V−AVF=BW and Sylvester matrix equations AV−EVF=BW,AV+BW=EVF and AV−VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.     

Representations for the weighted Drazin inverse of a sum of matrices

Three representations for the W-weighted Drazin inverse of a matrix A?CWB have been developed under some conditions where A,B,C∈? ^m×n , and W∈? ^n×m . The results of this paper not only extend the earlier works about the Drazin inverse and group inverse, but also weaken the assumed condition of a result of the Drazin inverse to the case where Γ _d ZZ _g =ZZ _g Γ _d is substituted with C(Γ _d ZZ _g ?ZZ _g Γ _d )B=0. Numerical examples are given to illustrate some new results.