On $k$-commutative matrices

n-fold Commutative Hyper K-ideals

 In this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper K-ideals. These definitions are thegeneralizations of the definitions of commutative, andimplicative hyper K-ideals, respectively, which have been definedin [12]. Then we obtain some related results. In particular wedetermine the relationships between n-fold implicative hyperK-ideal and n-fol...

Ela on K-potent Matrices

The paper provides extensive and systematic investigations of k-potent complex matrices, with a particular attention paid to tripotent matrices. Several new properties of k-potent matrices are identified. Furthermore, some results known in the literature are reestablished with simpler proofs than in the original sources and often in a generalized form.

A Theorem on Matrices over a Commutative Ring

A note on Potter’s theorem for quasi-commutative matrices

We discuss the converse of a theorem of Potter stating that if the matrix equation AB = ωBA is satisfied with ω a primitive qth root of unity, then Aq+Bq = (A+B)q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q = 3 and a conjecture for the general case. We also present som...

Non-commutative standard polynomials applied to matrices

The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra Mn(K). For K = R or C and 2≤ r ≤ 2n−1, we investigate how big Sr(A1, . . . ,Ar) can be when A1, . . . ,Ar belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is...