On a matrix group constructed from an {R,s+1,k}-potent matrix

Abstract

5For a {k}-involutory matrix R ∈ C n×n (that is, R k = I n ) and s ∈ {0, 1, 2, 3, . . . },this paper, a matrix group corresponding to a fixed {R, s + 1, k}-potent matrix is 8 explicitly constructed and properties of this group are derived and investigated. This 9 constructed group is then reconciled with the classical matrix group G A that is 10 associated with a generalized group invertible matrix A.

Spectral study of {R,s+1,k}- and {RCatral1, Lebtahi2, Stuart3 et al. 2020Journal of Computational and Applied Mathematics2020View full textAdd to dashboardCite

“…In recent years, investigations involving situations where a square matrix equals one of its powers (A s = A for some integer s ≥ 2; such a matrix is called {s}-potent) have been approached from both theoretical and applications points of view [4,7,8,14,19,20,24]. We mention just a selection of these studies here.…”

“…In [24,7], the authors generalized the concept of {s + 1}-potent matrices by means of the following definition. For a given {k}-involutory matrix R ∈ C n×n and a fixed positive integer s, a matrix A ∈ C n×n is called {R, s + 1, k}potent if satisfies RA = A s+1 R.…”

Matrices A such that A+1R = RA⁎ with R = I

“…For a k-involutory matrix R 2C n n and s 2 f0; 1; 2; 3; : : : g, a matrix A 2C n n is called fR; s + 1; kg-potent if A satis…es A s+1 R = RA [16,8]. These matrices generalize the centrosymmetric matrices (matrices A 2 C n n such that A = JAJ where J is the n n antidiagonal matrix [29]), the matrices A 2 C n n such that AP = P A where P is an n n permutation matrix [24], and fK; s+1gpotent matrices (matrices A 2 C n n for which KAK = A s+1 where K 2 = I n [17,18,19]).…”

“…In Section 3, we derive properties of fR; s + 1; k; g-potent matrices and give various characterizations. In [8] it was proved that an fR; s + 1; kgpotent matrix is always diagonalizable but this is not always true for matrices in P R;s;k; . We impose conditions on R or on the matrix A to recover some of the properties obtained for the former class of matrices.…”

“…While in [8] it was proved that an fR; s + 1; kg-potent matrix is always diagonalizable, this property is not always true for matrices in P R;s;k; . The next example illustrates this fact.…”