COMMON HERMITIAN LEAST-RANK SOLUTION OF MATRIX EQUATIONS $A_{1}X_{1}A_{1}^*=B_{1}$ AND $A_{2}X_{2}A_{2}^*=B_{2}$ SUBJECT TO INEQUALITY RESTRICTIONS

Sihem Guerarra, Said Guedjiba

DOI Number
-
First page
539
Last page
554

Abstract


In this paper, we establish a set of explicite formulas for calculating the maximal and minimal ranks and inertias of P-X with respect to X, where P∈ℂ_{H}ⁿ is given, X is a common Hermitian least-rank solution to matrix equations A₁XA₁^{∗}=B₁ and A₂XA₂^{∗}=B₂. As applications, we drive necessary and sufficient conditions for X≻P(≥P, ≺P, ≤P) in the löwner partial ordering. As consequence, we give necessary and sufficient conditions for the existence of common Hermitian positive (nonnegative, negative, nonpositive) definite least-rank solution to A₁XA₁^{∗}=B₁ and A₂XA₂^{∗}=B₂.

Keywords


Matrix equation, Rank formulas, Moore-Penrose generalized inverse, Hermitian, Least-rank solution, Inertia.

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References


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