In this paper, we establish some conditions for the existence and the representations for the common (P, Q) generalized reflexive and anti-reflexive solutions of matrix equations AX = B and XC = D, where P and Q are two generalized reflection matrices. Moreover, in corresponding solution set of the equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been presented.

a b s t r a c tIn this paper, a representation of (P + Q ) D such that PQP = 0, PQ 2 = 0 is given, which recovers the case PQ = 0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl., 322 (2001) 207-217]. Furthermore, we apply our results to establish the representations for the Drazin inverses of a 2 × 2 partitioned matrix M =  A B C D  , where A and D are square matrices. Finally, two numerical examples are given to illustrate our results.

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