Hermitian Nonnegative-definite Solution Research Articles

A General Hermitian Nonnegative-Definite Solution to the Matrix Equation <i>AXB</i> = <i>C</i>

We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.

Solvability of systems of linear matrix equations subject to a matrix inequality

In this paper, the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equationsand the Hermitian nonnegative definite solutions to the system of matrix equationsare, respectively, put forward, by making full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are, respectively, presented.

Common nonnegative definite solutions of some classical matrix equations

A necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of system of matrix equations $$A_{1}X=C_{1},\qquad XB_{2}=C_{2}, \qquad A_{3}XA_{3}^{\ast}=C_{3},\qquad A_{4}XA_{4}^{\ast}=C_{4} $$ as well as a representation for this general nonnegative definite solution are derived. As particular cases, the corresponding results on some other systems are also derived.

On Hermitian nonnegative-definite solutions to matrix equations

In this note, for a system of q matrix equations of the form $$ A_i XA_i^* = B_i B_i^* ,i = 1,2,...,q, $$ we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement.

On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA H =B

In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXA H =B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature.

The general common Hermitian nonnegative-definite solution to the matrix equations [formula omitted] and [formula omitted] with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA ∗=BB ∗ and CXC ∗=DD ∗ to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).

The general common Hermitian nonnegative-definite solution to the matrix equations AXA*=BB* and CXC*=DD* with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA ∗ =BB ∗ and CXC ∗ =DD ∗ to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).

The general common Hermitian Nonnegative-definite solution to the matrix equations AXA*=B and CXC*=D

We give a necessary and sufficient condition for the existence of at least one common Hermitian nonnegative-definite (positive-definite) solution to the pair of matrix equations AXA*=B and CXC*=D, and derive a representation of the general common Hermitian nonnegative-definite (positive-definite) solution to these two equations when they have at least such one common solution. This article can be viewed as a supplementary version of that derived by Young et al. (D.M. Young, J.W. Seaman Jr. and L.M. Meaux (1999). Independence distribution preserving covariance structures for themultivariate model. J. Multivariate Anal., 68, 165–175) since Groß (J. Groß (2000). Nonnegative-define and positive solutions to the matrix equation AXA*=B – revisited. Linear Algebra Appl., 321, 123–129) has given a counterexample to point out a mistake in their basic Theorem 1. The proposed approach is convenient to use and possesses good numerical reliability since it mainly involves only a series of singular value decompositions and the inverses of several positive definite diagonal matrices.

Nonnegative-definite and positive-definite solutions to the matrix equation [formula omitted] – revisited

The general Hermitian nonnegative-definite solution to the matrix equation AXA ∗= B is established in a form which can be viewed as a corrected version of that derived by C.G. Khatri and S.K. Mitra (SIAM J. Appl. Math. 31 (1976) 579–585) and an alternative version to that derived by J.K. Baksalary (Linear and Multilinear Algebra 16 (1984) 133–139). The new representation admits an easy way to obtain solutions of minimal and maximal rank, respectively.

A parametrization of the nonnegative definite solutions of the algebraic Riccati equation

In this paper, the set of the Hermitian nonnegative definite solutions of the algebraic Riccati equation is studied. The main result shows that any solution in this set is the product of a greatest solution and a projection. The analysis only assumes the existence of a greatest solution, and leads to generalize classification theorems via projections known in the stabilizable case.

Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness