The radius in matrix algebras—Examples and remarks

The main purpose of this note is to illustrate how the radius   in a finite-dimensional power-associative algebra over a field FF, either RR or CC, may change when the multiplication in this algebra is modified. Our point of departure will be Fn×nFn×n, the familiar algebra of n×nn×n matrices over FF with the usual matrix operations, where it is known that the radius is the classical spectral radius. We shall alter the multiplication in Fn×nFn×n in three different ways and compute, in each case, the radius in the resulting algebra.