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An $n \times n$ complex matrix $A$ is called Re-nonnegative definite (Re-nnd) if the real part of $x^{*} A x$ is nonnegative for every complex $n$ -vector $x$ . In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation $A X B = C$ are presented.

The symmetric linear matrix equation.

Ran, André C.M., Reurings, Martine C.B. (2002)

ELA. The Electronic Journal of Linear Algebra [electronic only]

The symmetric minimal rank solution of the matrix equation $A X = B$ and the optimal approximation.

Xiao, Qing-Feng, Hu, Xi-Yan, Zhang, Lei (2009)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain, Daniel Hershkowitz, Hans Schneider (1997)

Czechoslovak Mathematical Journal

Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $T V \to W$ is linear and $K \subset V$ and $D \subset W$ are cones, these results will be applied to $C = T (K)$ and $D$ , giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$ . The case when $V = W$ is the space $H$ of $n \times n$ Hermitian matrices, $D$ is the $n \times n$ positive semidefinite matrices, and $T (X) = A X + X^{*} A$ yields new and known results about the existence of block diagonal...

Trace inequalities for matrix products and trace bounds for the solution of the algebraic Riccati equations.

Liu, Jianzhou, Zhang, Juan, Liu, Yu (2009)

Journal of Inequalities and Applications [electronic only]

Two-step relaxation Newton method for nonsymmetric algebraic Riccati equations arising from transport theory.

Wu, Shulin, Huang, Chengming (2009)

Mathematical Problems in Engineering

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