Annals of Functional Analysis

On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations

Ying Li and Yongge Tian

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Abstract

Assume that the linear matrix equation $AXA^{*} = B = B^{*}$ has a Hermitian solution and is partitioned as $\left[\!\! \begin{array}{c} A_1 \\ A_2 \end{array} \!\!\right]\! X[\, A^{*}_1, \, A^{*}_2 \,] = \left[\!\! \begin{array}{cc} B_{11} \,, B_{12} \\ B_{21}^{*} \,, B_{22} \end{array}\right]$. We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations $A_1X_1A^{*}_1 = B_{11}$, $A_1X_2A^{*}_2 = B_{12}$ and $A_2X_3A^{*}_2 = B_{22}$. In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of $X-X_1-X_2-X_2^{*}-X_3$, and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality $X = X_1+X_2+X_2^{*}+X_3$ to hold and Hermitian matrix inequalities $X \geqslant(\leqslant, \cdots)\, X_1+X_2+X_2^{*}+X_3$ to hold in the Löwner partial ordering.

Article information

Source
Ann. Funct. Anal., Volume 5, Number 2 (2014), 30-46.

Dates
First available in Project Euclid: 7 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1396833500

Digital Object Identifier
doi:10.15352/afa/1396833500

Mathematical Reviews number (MathSciNet)
MR3192007

Zentralblatt MATH identifier
1296.15009

Subjects
Primary: 15A24: Matrix equations and identities
Secondary: 15B57: Hermitian, skew-Hermitian, and related matrices 47A62: Equations involving linear operators, with operator unknowns

Keywords
Matrix equation Hermitian solution equality inequality rank formula inertia formula

Citation

Li, Ying; Tian, Yongge. On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations. Ann. Funct. Anal. 5 (2014), no. 2, 30--46. doi:10.15352/afa/1396833500. https://projecteuclid.org/euclid.afa/1396833500


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