On relations among solutions of the Hermitian matrix equation \(AXA^* = B\) and its three small equations. (English) Zbl 1296.15009

Li, Ying; Tian, Yongge

Ann. Funct. Anal. 5, No. 2, 30-46 (2014).

Summary: Assume that the linear matrix equation \(AXA^\ast = B = B^\ast\) has a Hermitian solution and is partitioned as \(\left[\displaystyle{A_1\atop A_2}\right]X[A_1^\ast,A_2^\ast]=\left[\displaystyle{B_{11}\atop B_{21}^\ast}\;\displaystyle{B_{12}\atop B_{22}}\right]\). We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations \(A_1X_1A_1^\ast=B_{11}\), \(A_1X_2A_2^\ast=B_{12}\) and \(A_2X_3A_2^\ast=B_{22}\). In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of \(X-X_1-X_2-X_2^\ast-X_3\), and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality \(X=X_1+X_2+X_2^\ast+X_3\) to hold and Hermitian matrix inequalities \(X>(\geq,<,\leq) X_1+X_2+X_2^\ast+X_3\) to hold in the Löwner partial ordering.

MSC:

15A24	Matrix equations and identities
15B57	Hermitian, skew-Hermitian, and related matrices

Keywords:

matrix equation; Hermitian solution; equality; inequality; rank formula; inertia formula; Löwner partial ordering