Maximization and minimization of the rank and inertia of the Hermitian matrix expression with applications

✍ Scribed by Yongge Tian

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 465 KB
• Volume: 434
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.12.010

✦ Synopsis

We give in this paper a group of closed-form formulas for the maximal and minimal ranks and inertias of the linear Hermitian matrix function A -BX -(BX) * with respect to a variable matrix X. As applications, we derive the extremal ranks and inertias of the matrices X ±X * , where X is a solution to the matrix equation AXB = C, and then give necessary and sufficient conditions for the matrix equation AXB = C to have Hermitian, definite and Re-definite solutions.

In addition, we give closed-form formulas for the extremal ranks and inertias of the difference X 1 -X 2 , where X 1 and X 2 are Hermitian so-

then use the formulas to characterize relations between Hermitian solutions of the two equations.