The radius in matrix algebras--Examples and remarks

TL;DR

This paper explores how the radius in finite-dimensional power-associative algebras, especially matrix algebras, varies when the algebra's multiplication is modified, illustrating this with three different altered matrix algebra examples.

Contribution

It provides explicit computations of the radius in matrix algebras under three different non-standard multiplication modifications, highlighting how the radius can change.

Findings

Radius can differ from the classical spectral radius under modified multiplication.

Explicit examples of altered matrix algebras with computed radii.

Illustrates the dependence of the radius on algebraic structure modifications.

Abstract

The main purpose of this note is to illustrate how the radius in a finite-dimensional power-associative algebra over a field $\mathbb{F}$, either $\mathbb{R}$ or $\mathbb{C}$, may change when the multiplication in this algebra is modified. Our point of departure will be $\mathbb{F}^{n \times n}$, the familiar algebra of $n \times n$ matrices over $\mathbb{F}$ with the usual matrix operations, where it is known that the radius is the classical spectral radius. We shall alter the multiplication in $\mathbb{F}^{n \times n}$ in three different ways and compute, in each case, the radius in the resulting algebra.