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Extreme ranks of real matrices in solution of the quaternion matrix equation $AXB=C$ with applications. (English) Zbl 1188.15016
Summary: For a consistent quaternion matrix equation $AXB=C$, the formulas are established for maximal and minimal ranks of real matrices ${X}_{1},{X}_{2},{X}_{3},{X}_{4}$ in solution $X={X}_{1}+{X}_{2}i+{X}_{3}j+{X}_{4}k$. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices $E,F,G,H$ in a generalized inverse ${\left(A+Bi+Cj+Dk\right)}^{-}=E+Fi+Gj+Hk$ of a quaternion matrix $A+Bi+Cj+Dk$ are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations ${A}_{1}X{B}_{1}={C}_{1}$ and ${A}_{2}X{B}_{2}={C}_{2}$ to have a common real solution.
##### MSC:
 15A24 Matrix equations and identities 15A33 Matrices over special rings 15A03 Vector spaces, linear dependence, rank 15A09 Matrix inversion, generalized inverses