## A Hermitian least squares solution of the matrix equation $$AXB=C$$ subject to inequality restrictions.(English)Zbl 1268.15012

Summary: We give some closed-form formulas for computing the maximal and minimal ranks and inertias of $$P-X$$ with respect to $$X$$, where $$P\in \mathbb{C}_H^n$$ is given, and $$X$$ is a Hermitian least squares solution to the matrix equation $$AXB=C$$. We derive, as applications, necessary and sufficient conditions for $$X\geq$$ $$(\leq,>,<)P$$, in the Löwner partial ordering. In addition, we give necessary and sufficient conditions for the existence of a Hermitian positive (negative, nonpositive, nonnegative) definite least squares solution to $$AXB=C$$.

### MSC:

 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices
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### References:

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