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Minimal ranks of some quaternion matrix expressions with applications. (English) Zbl 1204.15005
Let p(X,Y)=A-BX-X (*) B (*) -CYC (*) and q(X,Y)=A-BX+X (*) B (*) -CYC (*) be quaternion matrix expressions, where A is persymmetric or perskew-symmetric. The authors derive the minimal rank formula of p(X,Y) with respect to pair of matrices X and Y=Y (*) , and the minimal rank formula of q(X,Y) with respect to pair of matrices X and Y=-Y (*) . As applications, they establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.
15A03Vector spaces, linear dependence, rank
15A33Matrices over special rings
15A24Matrix equations and identities
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