On the group inverse of linear combinations of two group invertible matrices

Ela on the Group Inverse of Linear Combinations of Two Group Invertible Matrices

hold. If such matrix X exists, then it is unique, denoted by A, and called the group inverse of A. It is well known that the group inverse of a square matrix A exists if and only if rank(A) = rank(A) (see, for example, [1, Section 4.4] for details). Clearly, not every matrix is group invertible. It is straightforward to prove that A is group invertible if and only if A is group invertible, and ...

Group Inverse and Generalized Drazin Inverse of Block Matrices in a Banach Algebra

Let A be a complex unital Banach algebra with unit 1. The sets of all invertible and quasinilpotent elements (σ(a) = {0}) of A will be denoted by A and A, respectively. The group inverse of a ∈ A is the unique element a ∈ A which satisfies aaa = a, aaa = a, aa = aa. If the group inverse of a exists, a is group invertible. Denote by A the set of all group invertible elements of A. The generalize...

A brief introduction to quaternion matrices and linear algebra and on bounded groups of quaternion matrices

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

On the nonnegative inverse eigenvalue problem of traditional matrices

In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.

On the denseness of the invertible group in uniform Frechet algebras