When is the hermitian/skew-hermitian part of a matrix a potent matrix?

Abstract

Abstract. This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form X Y * Y 0 , and a singular value decomposition of A. In addition, a relation between the potency of H(A), respectively S(A), and the normality of A is also studied.

On k-potent matrices

“…Quadripotent matrices recently focused also some special interest, which originates mostly from the fact that they occur naturally in considerations dealing with generalized and hypergeneralized projectors introduced in [20]. In addition to the papers [25], [27], [30], [31], and [43], each of which contains a systematical study over a selected topic concerning k-potent matrices, a collection of related isolated results was published in recent years in a number of independent articles. Many of these results are recalled in the present paper, in which k-potent matrices are revisited and extensively investigated.…”

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.…”

The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices

“…Collections of results dealing with idempotent and tripotent matrices are available in several monographs emphasizing their usefulness in statistics, for instance [7] (Section 12.4), [8] (Chapter 7), and [9] (Sections 8.6, 8.7, and 20.5.3). In addition to the papers [10][11][12][13][14][15][16][17][18][19][20], each of which contains a systematic study over a selected topic concerning k-potent matrices, a collection of related isolated results was published in recent years in a number of independent articles. Apart from the papers mentioned above, an inspiration for this paper was also the work of Baksalary et al [21], where the authors discussed the nonsingularity of a linear combination of two idempotent matrices, and the work of Koliha et al [22], where the authors considered the nullity and rank of linear combinations of two idempotent matrices.…”