Representations for the Drazin inverses of the sum of two matrices and some block matrices

On the Group Inverse for the Sum of Matrices

“…Applying Lemma 2.1, the group inverse of P + Q = U ∆ Q 2 Q 3 Q 4 U −1 exists if and only if G = ∆ 2 + Q 2 S π Q 3 is [4] On the group inverse for the sum of matrices invertible. Applying Lemma 2.1, the group inverse of P + Q = U ∆ Q 2 Q 3 Q 4 U −1 exists if and only if G = ∆ 2 + Q 2 S π Q 3 is [4] On the group inverse for the sum of matrices invertible.…”

Some additive results for the Drazin inverse and its application

“…(i) P 2 Q = 0, Q 2 = 0 (see [6]); (ii) P 2 Q = 0, Q 2 P = 0 (see [7]); (iii) QP 2 = 0, Q 2 = 0 (see [6]); (iv) PQ 2 = 0, QP 2 = 0 (see [7]).…”

“…In some papers the expression of M D is given under conditions which concern the generalized Schur complement of matrix M defined by S = D − CA D B. Here we list some of them: (i) CA π = 0, A π B = 0, and S = 0 (see [9]); (ii) CA π B = 0, AA π B = 0, and S = 0 (see [3]); (iii) CA π B = 0, CAA π = 0, and S = 0 (see [3]); (iv) ABCA π = 0, BCA π is nipotent, and S = 0 (see [6]); (v) A π BCA = 0, A π BC is nipotent, and S = 0 (see [6]); (vi) ABCA π = 0, A π ABC = 0, and S = 0 (see [7]); (vii) ABCA π = 0, CBCA π = 0, and S = 0 (see [7]); (viii) ABCA π A = 0, ABCA π B = 0, and S = 0 (see [11]); (ix) AA π BCA = 0, CA π BCA = 0, and S = 0 (see [11]). In this paper, we derive a new representation for M D under the conditions A π ABC = 0, A 2 BCA π A = 0, A 2 BCA π B = 0 and S = 0.…”

Additive and Product Properties of Drazin Inverses of Elements in a Ring

“…For two commutative Drazin invertible elements a, b ∈ R,Zhuang, Chen et al [16] proved that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible.Moreover, the representation of (a + b) D was obtained. More results on Drazin inverse can be found in [1][2][3][4][5][6]8,[11][12][13][14][15][16].…”