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In this paper, weuse the well-known Lowner order and the core partial order to introduce a new partial order CL <= on the class of core matrices which is not dominated by any of the knownmatrix partial orders. We characterize CL <...
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In this paper, weuse the well-known Lowner order and the core partial order to introduce a new partial order CL <= on the class of core matrices which is not dominated by any of the knownmatrix partial orders. We characterize CL <=, study its relations with the Lowner partial order under constraints, and exemplify its differences with other partial orders.
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The purpose of this paper is to develop inequalities for Kronecker products and Hadamard products ofpositive definite matrices. A number of inequalities involving powers, Kronecker powers, and Hadamard powers of linearcombination ...
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The purpose of this paper is to develop inequalities for Kronecker products and Hadamard products ofpositive definite matrices. A number of inequalities involving powers, Kronecker powers, and Hadamard powers of linearcombination of matrices are presented. In particular, Holder inequalities and arithmetic mean-geometric mean inequalitiesfor Kronecker products and Hadamard products are obtained as special cases.
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Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A(2) and B...
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Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A(2) and B-2, in the sense of the star partial ordering. the minus partial ordering, and the Lowner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. (C) 2003 Elsevier Inc. All rights reserved. [References: 12]
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Gross [Linear Algebra Appl. 326 (2001) 215] developed characterizations of the minus and star partial orders between the squares of Hermitian nonnegative definite matrices referring to the concept of the space preordering. In the ...
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Gross [Linear Algebra Appl. 326 (2001) 215] developed characterizations of the minus and star partial orders between the squares of Hermitian nonnegative definite matrices referring to the concept of the space preordering. In the present paper, his results are generalized by deleting the nonnegative definiteness assumption and supplemented by alternative characterizations. (C) 2003 Elsevier Inc. All rights reserved.
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In this paper, we first investigate the matrix equation AXB + CYD = G in a new fashion, where A, B, C, D and G are arbitrary matrices. Then, we establish some necessary and sufficient conditions for the existence of a solution of ...
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In this paper, we first investigate the matrix equation AXB + CYD = G in a new fashion, where A, B, C, D and G are arbitrary matrices. Then, we establish some necessary and sufficient conditions for the existence of a solution of AXA * + BX * B * = C, where A, B are arbitrary matrices and C is a Hermitian matrix. In the special case when B = A, we determine the general solution of A(X+ X *) A * = C, where A is an arbitrary matrix and C is a Hermitian matrix.
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In this paper, we first establish some necessary and sufficient conditions for the existence of Hermitian solution of AXA* = B subject to CXC* > D, where B and D are Hermitian matrices. Furthermore, a general expression for this H...
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In this paper, we first establish some necessary and sufficient conditions for the existence of Hermitian solution of AXA* = B subject to CXC* > D, where B and D are Hermitian matrices. Furthermore, a general expression for this Hermitian constrained solution is derived, several special cases are also considered. (C) 2015 Elsevier Inc. All rights reserved.
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Assume that a quadratic matrix-valued function is given and let be the set of all least-squares solutions of the linear matrix equation . In this paper, we first establish explicit formulas for calculating the maximum and minimum ...
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Assume that a quadratic matrix-valued function is given and let be the set of all least-squares solutions of the linear matrix equation . In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of subject to , and then derive from the formulas the analytic solutions of the two optimization problems and subject to in the Lowner partial ordering. As applications, we present a variety of results on equalities and inequalities of the ordinary least squares estimators of unknown parameter vectors in general linear models.
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Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator By represents the best linear unbiased estimator (BLUE) of the expectation vector Xbeta un...
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Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator By represents the best linear unbiased estimator (BLUE) of the expectation vector Xbeta under the general Gauss-Markov model M = {y, Xbeta, sigma(2)V} if and only if B(X : VXperpendicular to) = (X : 0), where X-L is any matrix whose columns span the orthogonal complement to the column space of X. In this note, still another development of such a characterization is proposed with reference to the BLUE of any vector of estimable parametric functions Kbeta. From the algebraic point of view, the present development seems to be the simplest from among all accessible in the literature till now. (C) 2003 Elsevier Inc. All rights reserved.
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Matrices with the property that the real part is positive definite, have been studied for the past five decades or more. Many results in the form of inequalities have been obtained for matrices possessing this property. In this ar...
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Matrices with the property that the real part is positive definite, have been studied for the past five decades or more. Many results in the form of inequalities have been obtained for matrices possessing this property. In this article, a new class of matrices, viz., matrices whose real part is positive semidefinite, is considered, wherein extensions of the results in the literature are obtained.
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The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. If we take the inertia and rank of a Herm...
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The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. If we take the inertia and rank of a Hermitian matrix as objective functions, then they are neither differentiable nor smooth. In this case, maximizing and minimizing the inertia and rank of a Hermitian matrix function could be regarded as a continuous-integer optimization problem. In this paper, we use some pure algebraic operations of matrices and their generalized inverses to derive explicit expansion formulas for calculating the global maximum and minimum ranks and inertias of the linear Hermitian matrix function A + BXB~* subject to some rank and definiteness restrictions on the variable matrix X. Various direct consequences of the formulas in characterizing algebraic properties of A + BXB~* are also presented. In particular, solutions to a group of constrained optimization problems on the rank and inertia of a partially specified block Hermitian matrix are given.
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