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Characterizations of the group invertibility of a matrix revisited

Tian Yongge

Demonstratio Mathematica

2022

Vol. 55, nr 1

866--890

A square complex matrix A is said to be group invertible if there exists a matrix X such that AXA=A, XAX=X, and AX=XA hold, and such a matrix X is called the group inverse of A . The group invertibility of a matrix is one of the fundamental concepts in the theory of generalized inverses, while group inverses of matrices have many essential applications in matrix theory and other disciplines. The purpose of this article is to reconsider the characterization problem of the group invertibility of a matrix, as well as the constructions of various algebraic equalities in relation to group invertible matrices. The coverage includes collecting and establishing a family of existing and new necessary and sufficient conditions for a matrix to be group invertible and giving many algebraic matrix equalities that involve Moore-Penrose inverses and group inverses of matrices through the skillful use of a series of highly selective formulas and facts about ranks, ranges, and generalized inverses of matrices, as well as block matrix operations.

Application of the partitioning method to specific Toeplitz matrices

Stanimirović P., Miladinović M., Stojanović I., Miljković S.

International Journal of Applied Mathematics and Computer Science

2013

Vol. 23, no. 4

809--821

We propose an adaptation of the partitioning method for determination of the Moore–Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore–Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in MATLAB, and illustrative examples are presented.

Throught a generalized inverse

Stępniak C.

Demonstratio Mathematica

2008

Vol. 41, nr 2

291-296

Traditionally, the existence of a generalized inverse of a matrix A is derived in an indirect way from the matrix equation AXA = A. We reach this result in a direct and constructive manner, based on spectral decomposition. Moreover, some new results on its characterization and on representation of the entire set of generalized inverses are given. Usefulness of these results is demonstrated in examples.