On k-idempotent 0-1 matrices

Abstract

Let k ≥ 2 be an integer. If a square 0-1 matrix A satisfies A k = A, then A is said to be k-idempotent. In this paper, we give a characterization of k-idempotent 0-1 matrices. We also determine the maximum number of nonzero entries in kidempotent 0-1 matrices of a given order as well as the k-idempotent 0-1 matrices attaining this maximum number.

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

“…Proof. Let A, B, C ∈ C n×n be of the form (1), (15), and (19), respectively, and r(A) = r. Now, (17) and (23), the form (25) follows immediately.…”

“…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.…”

Sums Of K-potent Matrices

“…Let k ≥ 2 be an integer. If a square matrix A satisfies A k = A, then A is said to be k-potent (see definition in [4]). For k = 2, A is said idempotent.…”