Group inverses of matrices with path graphs

Abstract

Abstract.A simple formula for the group inverse of a 2 × 2 block matrix with a bipartite digraph is given in terms of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of such matrices are given. An extension of the graph-theoretic description of the group inverse to singular matrices with tree graphs is conjectured.

Block representations of the Drazin inverse of a bipartite matrix

“…Note that rank CB = 3 > rank CBC = 2 and index CB = 2; thus by Theorem 3.6 or Corollary 3.7, index A = 2s + 1 = 3. Although A # does not exist, (BC) # does exist and thus results in [5] can be applied to determine A D in this example.…”

Some results on the group inverse for block matrices over skew fields

“…Indeed, generalized inverses are useful tools in areas such as special matrix theory, singular differential and difference equations and graph theory; see [5], [9] [11], [12] and [15]. For example, in [9] it is shown that the adjacency matrix of a bipartite graph can be written in the form of 0 B C 0 , and necessary and sufficient conditions are given for the existence and representation of the group inverse of a block matrix 0 B C 0 .…”

A note on block representations of the group inverse of Laplacian matrices

“…For a matrix B, let B + denote the Moore-Penrose inverse of B. Some representations for the group inverse of block matrices (operators) are given in [2][3][4][5][6][7][8]11,12,14,16,17,28]. More details for the theory of generalized inverse can be found in [9].…”