Some matrix equations with applications†

Abstract

We establish necessary and sufficient conditions for the solvability to the matrix equation

and present an expression of the general solution to (1) when it is solvable. As applications, we discuss the consistence of the matrix equation

where * means conjugate transpose, and provide an explicit expression of the general solution to (2). We also study the extremal ranks of X ₃ and X ₄ and extremal inertias of and in (1). In addition, we obtain necessary and sufficient conditions for the classical matrix equation

to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite and Re-negative definite solutions. The findings of this article extend related known results.

†Dedicated to Professor Ky Fan (1914–2010).

Keywords:

• matrix equation
• Moore–Penrose inverse
• rank
• inertia
• re-nonnegative semidefinite matrix

AMS Subject Classifications::

• 15A24
• 15A09
• 15A03

Acknowledgements

The authors thank the anonymous referees for their valuable suggestions that improved the exposition of this article. This research was supported by the grants from the National Natural Science Foundation of China (11171205), Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001) and Shanghai Leading Academic Discipline Project (J50101).

Notes

†Dedicated to Professor Ky Fan (1914–2010).