Abstract
Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.
Highlights
Let Cm×n denote the set of all m × n complex matrices
When ind(A) = 1, the matrix AD is called the group inverse of A and it is denoted by A#
We give representations of the Drazin inverse of 2 × 2 block matrix M under some conditions relating to the Schur complement of M, which generalized the results studied by Cvetkovic-Ilic [9]
Summary
Let Cm×n denote the set of all m × n complex matrices. We use R(A), N(A), and r(A) to denote the range, the null space, and the rank of a matrix A, respectively. We give representations of the Drazin inverse of 2 × 2 block matrix M under some conditions relating to the Schur complement of M, which generalized the results studied by Cvetkovic-Ilic [9]. Baksalary and Styan [10] presented the necessary and sufficient conditions under which {1}, {2}, {1, 2}, {1, 3}, and {1, 4}-inverses of block matrix M can be represented by the Banachiewicz-Schur form. Cvetkovic-Ilic [9] gave necessary and sufficient conditions which ensure the representation of the MP-inverse of a block matrix M by both of the Banachiewicz-Schur forms. M{1, 2, 3}, by Lemmas 4 and 6, we get that, for arbitrary A− ∈ A{1, 2, 3}, D− ∈ D{1, 2, 3}, and G− ∈ G{1, 2, 3}, the following hold: FAB = 0, FGB = 0, and FDC = 0.
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