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Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications. (English) Zbl 1227.15015

Authors’ abstract: We establish the formulas of the maximal and minimal ranks of a 3×3 partial banded block matrix

M 11 M 12 XM 21 M 22 M 23 YM 32 M 33

where X and Y are a pair of variant quaternion matrices subject to linear quaternion matrix equations A 1 X=C 1 , XB 1 =C 2 , A 2 Y=D 1 , YB 2 =D 2 . As applications, we present a necessary and sufficient condition for the solvability to the quadratic system A 1 X=C 1 , XB 1 =C 2 , A 2 Y=D 1 , YB 2 =D 2 , XPY=J over the quaternion algebra. We also give the conditions for the rank invariance of the quadratic matrix expression XPY=J subject to the linear quaternion matrix equations mentioned above.

MSC:
15A24Matrix equations and identities
15A33Matrices over special rings
15A03Vector spaces, linear dependence, rank
15A09Matrix inversion, generalized inverses