Extremal Ranks of Some Symmetric Matrix Expressions with Applications
Yongge Tian, Yonghui Liu · SIAM Journal on Matrix Analysis and Applications · 2006
Suppose $A - BXB^*$, $A - BX - X^*B^*$, and $A - BX + X^*B^*$ are three linear matrix expressions over the field of complex numbers, where A is an Hermitian or skew‐Hermitian matrix. In this paper, we consider how to choose an Hermitian or skew‐Hermitian matrix X such that $A - BXB^*$ have the maximal and minimal possible ranks, and how to choose X such that $A - BX \pm X^*B^*$ attain the minimal possible ranks. Some applications to Hermitian or skew‐Hermitian solutions of matrix equations with symmetric patterns are also given.