Hai-Ying Shan

Matrices with signed generalized inverses

Abstract A real matrix A is said to have a signed generalized inverse , if the sign pattern of its generalized inverse A + is uniquely determined by the sign pattern of A . In this paper, we give complete characterizations of the m×n matrices A which have signed generalized inverses, for both cases ρ(A)=n and ρ(A) (where ρ(A) is the term rank of A , and without loss of generality we assume n⩽m ), and thus solve a problem proposed in [B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056]. Using these characterizations, we are also able to show that the property of having a signed generalized inverse for a matrix A is inherited by all the submatrices B of A with ρ(B)=ρ(A) and is also inherited by all those matrices A 1 with ρ(A 1 )=ρ(A) which can be obtained from A by replacing some nonzero entries of A by zero. We also consider several special cases of a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, MA, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterization of the matrices in a special triangular block form to have signed generalized inverses.

The solution of a problem on matrices having signed generalized inverses

Abstract A real matrix A is said to have a signed generalized inverse , if the sign pattern of its generalized inverse A + is uniquely determined by the sign pattern of A . In this paper, we solve a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterizations of the matrices with a special lower triangular blocked form to have a signed generalized inverse. Using this characterization and the fact that every matrix which has a signed generalized inverse and full column term rank and no zero rows is permutation equivalent to a matrix of this form, we give two algorithms for determining whether or not a matrix with full column term rank, or a general matrix, has a signed generalized inverse.

Matrices with Special Sign Patterns of Signed Generalized Inverses

A real matrix A is said to have a signed generalized inverse (GI) if the sign pattern of its GI A+ is uniquely determined by the sign pattern of A. We characterize those sign pattern matrices with a signed GI, and the GI of it is nonnegative, or is positive, or has no zeros.

Matrices with totally signed powers

A square real matrix A is said to have signed d-power, if the sign pattern of the power Ad is uniquely determined by the sign pattern of A. A is said to have totally signed powers if A has signed d-powers for all positive integers d. A is said to be d-powerful if all the non-zero terms in the expansion formula of each entry of Ad have the same sign. A is powerful if A is d-powerful for all positive integers d. We show that A has totally signed powers is equivalent to A being powerful, although A has signed d-power is not equivalent to A being d-powerful.

Number of nonzero entries of S2NS matrices and matrices with signed generalized inverses

Abstract A real matrix A has a signed generalized inverse (or signed GI), if the sign pattern of its generalized inverse A + is uniquely determined by the sign pattern of A . The notion of matrices having signed GI’s is a generalization of the well known notion of strong SNS matrices (or S 2 NS matrices). Sharp bounds, and characterization of equality, for the number of nonzero entries of S 2 NS matrices of order n are given. Then sharp bounds, and characterization of equality, for the number of nonzero entries of m × n matrices with signed GI’s are given.