On Commutative Matrices

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...

Inversion of Matrices over a Commutative Semiring

It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative sem...

Delayed Feedback Control by Commutative Gain Matrices

The delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spect...