Ela Group Inverse for the Block Matrix with Two Identical Subblocks over Skew Fields
نویسندگان
چکیده
is called the Drazin inverse of A and is denoted by X = A, where k is the index of A, i.e., the smallest non-negative integer such that rank(A) = rank(A). We denote such a k by Ind(A). It is well-known that A exists and is unique (see [2]). If Ind(A) = 1, A is also called the group inverse of A and is denoted by A. Then A exists if and only if rank(A) = rank(A) (see [1, 3, 11-14, 24, 25, 29]). We denote I −AA♯ by A. The group inverse of block matrices has numerous applications in matrix theory, such as singular differential and difference equations, Markov chains, iterative methods and so on (see [12]-[14]). For instance, Y. Wei et al. studied the representation of the group inverse of a real singular Toeplitz matrix which arises in scientific computing and engineering (see [29]); in [25], S. Kirkland et al. investigated the representation of the group inverse of the Laplacian matrix of an undirected weighted graph G on n vertices; and in [24], G. Heinig studied the group inverse of the Sylvester transformation φ(X) = AX −XB, where A ∈ C and B ∈ C; utilizing the theory of Drazin inverse, the differential equation Ax + Bx = f is studied in linear systems,
منابع مشابه
Block Matrices over Skew Fields
In this paper, we give the existence and the representation of the group inverse for circulant block matrix M = ( A B B A ) (A, B ∈ Kn×n, andA = A, B = B) over skew field . Some relative additive results are also given. Mathematics Subject Classification: 47H09; 47H10
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