Search

Let $ p_1 \lt p_2 \lt \cdots \lt p_n $ p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function $ x^{r+1} $ x r + 1 given by $ L_{r+1}=\begin {bmatrix}\frac {p_i^{r+1}-p_j^{r+1}}{p_i-p_j}\end {bmatrix} $ L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix $ P_r=[\begin {smallmatrix}{(p_i+p_j)^r}\end {smallmatrix}] $ P r = [ (p i + p j) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between $ L_{r+1} $ L r + 1 and $ P_r $ P r . Indeed, a non-singular matrix X over $ \mathbb {R} $ R is explicitly obtained such that $ X'P_rX=L_{r+1} $ X ′ P r X = L r + 1 in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the m -core inverse in the Minkowski space, and get a sufficient and necessary condition for the existence of the inverse and some other related properties. Furthermore, by using the inverse, we introduce the m -core partial ordering and obtain solutions (or restricted least squares solutions) of some matrix equations in the Minkowski space. [ABSTRACT FROM AUTHOR]

This paper deals with the idempotency, involution and nilpotency of two matrices. In this paper, by introducing a new idea into the classical block technique, we characterize all situations in which a linear combination of the formis(Q1) square-zero when P and Q are both idempotent;(Q2) square-zero when P is tripotent and Q is idempotent, respectively;(Q3) idempotent when P is tripotent and Q is involutory, respectively;(Q4) involutory when P is tripotent and Q is involutory, respectively.In addition, we point out several other related problems that the reformed method in this paper applies. [ABSTRACT FROM AUTHOR]

Conventional distributed beamforming schemes in multicell system often ignore two practical conditions, (1) the coexistence of intracell interference (InCI) and inter-cell interference (ICI), (2) the number of users is larger than the number of base station (BS) antennas. In this paper, we consider these two conditions and analyze the decentralized beamforming of the multicell system from the viewpoint of zero forcing (ZF). A novel user selection method based on ZF algorithm is proposed to divide the users into two categories, i.e. one of the categories is eligible to participate in ZF algorithm and the other is not. The proposed criterion of partitioning users is based on the degree of parallelity between local channels. In addition, each BS utilizes only the local channel state informations (CSIs) to derive its beamforming vector. The simulation results demonstrate that our proposed algorithm can achieve higher rates and are more robust against InCI and ICI than the traditional ones under the different number of transmitting antennas and users. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the EP-nilpotent decomposition and present some of its applications. By applying the decomposition, we introduce two partial orders (the E-N partial order and the E-S partial order) and characterize their properties. The two partial orders above are non-minus-type partial orders, although the method used to introduce the two partial orders is similar to that of the C-N partial order, a minus-type partial order. [ABSTRACT FROM AUTHOR]

In this paper, we consider an expression of the general solution to the classical system of matrix equationsWe present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the systemand the systemwheremeans conjugate transpose. We also give the expressions of the general solutions to the systems. [ABSTRACT FROM AUTHOR]

In this paper, we use the well-known Löwner order and the core partial order to introduce a new partial orderon the class of core matrices which is not dominated by any of the known matrix partial orders. We characterize, study its relations with the Löwner partial order under constraints, and exemplify its differences with other partial orders. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative method is presented for finding the bisymmetric solutions of a pair of consistent matrix equations A1XB1=C1, A2XB2=C2, by which a bisymmetric solution can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the solution with least Frobenius norm can be obtained by choosing a special kind of initial matrix. In the solution set of the matrix equations, the optimal approximation bisymmetric solution to a given matrix can also be derived by this iterative method. The efficiency of the proposed algorithm is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following minimization problem:where,,,andare given. An efficient inequality relaxation technique is presented to relax the matrix inequality constraint so that there is an optimal solution which is(R,S)-symmetric that minimize, and also satisfies the corrected matrix inequality constraint. A hybrid algorithm with convergence analysis is given to solve this problem. Numerical examples show that the algorithm requires less CPU times when compared with some other methods. [ABSTRACT FROM AUTHOR]

Euclidean distance matrices ( EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ( GEDM s) that include EDM s. Each GEDM is an entry-wise nonnegative matrix. A GEDM is not symmetric unless it is an EDM. By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore–Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices. [ABSTRACT FROM AUTHOR]

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix G (λ) when G (λ) has the same structure. We also describe recovery of eigenvectors, minimal bases and minimal indices of G (λ) from those of the linearizations of G (λ) and show that the recovery is operation-free. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are obtained for a pair of matrix equations AX = C and XB = D to have common Re-nnd and Re-pd solutions and the explicit representations of the general common Re-nnd and Re-pd solutions are given when the stated conditions hold. [ABSTRACT FROM AUTHOR]

In this note, we prove that ifarecontractive matrices andwith, then for each. This generalizes an inequality given by Marcus in 1958. [ABSTRACT FROM PUBLISHER]

Letandwe first solve the minimum Frobenius norm residual problem (Problem LSP):with unknown Toeplitz matricesXandY. We then consider a best approximation problem: given Toeplitz matricesand, findsuch thatwhereis the solution set of Problem LSP. We show that the best approximation solutionis unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

Letbe thealgebra ofcomplex matrices and letbe a completely positive map. Supposeis a self-adoint matrix. We prove a submajorization result concerning positive and negative parts of the spectrum of. As a consequence, we obtain inequalities concerning the smallest and the largest eigenvalues of the Schur product ofand, whereandare self-adjoint. [ABSTRACT FROM PUBLISHER]

We study those pairs of subspaces of a complex Hilbert space with the same angle and minimal angle and present several characterizations of the pairs of subspaces with angle equal to. [ABSTRACT FROM PUBLISHER]

Let g, h:V×V→ be two non-degenerate symmetric bilinear forms on a finite-dimensional complex vector space V. Let G (resp. H) be the Lie group of isometries of g (resp. h). If the endomorphism L:V→V associated to g, h is diagonalizable, then the dimension of the intersection group G∩H is computed in terms of the dimensions of the eigenspaces of L. [ABSTRACT FROM AUTHOR]