Extremal Ranks of Some Symmetric Matrix Expressions with Applications

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.

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