Changjiang Bu

Group inverse for a class 2×2 block matrices over skew fields

Abstract Suppose K is a skew field. Let K n × n denote the set of all matrices over K. For A ∈ K n × n , the matrix X ∈ K n × n is said to be the group inverse of A, if it holds that AXA = A , XAX = X , AX = XA . In this paper, we give the existence and the representation of the group inverse for block matrix M = A A B 0 ( A , B ∈ K n × n , A 2 = A ) over skew fields, and give the existences and the representations of the group inverse for other block matrices.

Some results on resistance distances and resistance matrices

In this paper, we obtain formulas for resistance distances and Kirchhoff index of subdivision graphs. An application of resistance distances to the bipartiteness of graphs is given. We also give an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix.

Group inverse for the block matrices with an invertible subblock

Let M=ABCD (A is square) be a square block matrix with an invertible subblock over a skew field K. In this paper, we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for M under some conditions.

Group inverse for block matrices and some related sign analysis

The sign pattern of a real matrix M is the (0, 1, −1)-matrix obtained from M by replacing each entry by its sign. Let N ∈ ℝ n×n be a group invertible matrix. Let Q(N) be the set of real matrices with the same sign pattern as N. For any , if is group invertible and the group inverses of N and have the same sign pattern, then N is called an S2GI matrix. In this article, we present the existence and the representations for the group inverse of some block matrices with one or two full rank sub-blocks, and give a family of block matrices which are S2GI matrices. Applying these results, we can partially determine the sign pattern of the solution of singular linear system with index one.

Representations of the Drazin inverse on solution of a class singular differential equations

In 1983, Campbell proposed a problem to find an explicit representation of the Drazin inverse for the block matrix with E and F square in terms of E and F according to the research on singular differential equations. In this article, we give the representations of the Drazin inverse for with E and F square under the condition that EF = FE and the Drazin inverse for with E square under the condition that EFG = FGE, and then give some representations under some other conditions, so as to solve the problem partially and to get some extra results.

Moore–Penrose inverse of tensors via Einstein product

In this paper, we define the Moore–Penrose inverse of tensors with the Einstein product, and the explicit formulas of the Moore–Penrose inverse of some block tensors are obtained. The general solutions of some multilinear systems are given and we also give the minimum-norm least-square solution of some multilinear systems using the Moore–Penrose inverse of tensors.

Some results on the group inverse of the block matrix with a sub-block of linear combination or product combination of matrices over skew fields

Let K be any skew field and K m×n be the set of all the m × n matrices over K. In this article, necessary and sufficient conditions are given for the existence of the group inverses of block matrix , where and exist, A = c 1 B + c 2 C, non-zero elements c 1 and c 2 are in the centre of K and block matrix , where A = B k C l , k and l are positive integers. Then the representations of the group inverses of these block matrices are also given.

A NOTE ON BLOCK REPRESENTATIONS OF THE GROUP INVERSE OF LAPLACIAN MATRICES

Let G be a weighted graph with Laplacian matrix L and signless Laplacian matrix Q. In this note, block representations for the group inverse of L and Q are given. The resistance distance in a graph can be obtained from the block representation of the group inverse of L.

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

Abstract In this paper, we give some formulas of the Drazin inverses of the sum of two matrices under the conditions P 2 Q = 0 , Q 2 P = 0 and P 3 Q = 0 , QPQ = 0 , QP 2 Q = 0 respectively. And we also give some representations for the Drazin inverse of block matrix A B C D ( A and D are square) under some conditions.

Spectral characterization of line graphs of starlike trees

Two graphs are said to be A-cospectral if they have the same adjacency spectrum. A graph G is said to be determined by its adjacency spectrum if there is no other non-isomorphic graph A-cospectral with G. A tree is called starlike if it has exactly one vertex of degree greater than 2. In this article, we prove that the line graphs of starlike trees with maximum degree at least 12 are determined by their adjacency spectra.