0000000000546178
AUTHOR
M. Catral
On a matrix group constructed from an {R,s+1,k}-potent matrix
Let R is an element of C-nxn be a {k}-involutory matrix (that is, R-k = I-n) for some integer k >= 2, and let s be a nonnegative integer. A matrix A is an element of C-nxn is called an {R,s + 1, k}-potent matrix if A satisfies RA = A(s+1)R. In this paper, a matrix group corresponding to a fixed {R,s + 1, k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group G(A) that is associated with a generalized group invertible matrix A.
Spectral study of {R, s + 1, k}- and {R, s + 1, k, *}-potent matrices
[EN] The {R, s +1, k}- and {R, s +1, k, *}-potent matrices have been studied in several recent papers. We continue these investigations from a spectral point of view. Specifically, a spectral study of {R, s + 1, k} -potent matrices is developed using characterizations involving an associated matrix pencil (A, R). The corresponding spectral study for {R, s+ 1, k, *}-potent matrices involves the pencil (A*, R). In order to present some properties, the relevance of the projector I - AA(#). where A(#) is the group inverse of A is highlighted. In addition, some applications and numerical examples are given, particularly involving Pauli matrices and the quaternions.