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 $\mathbb{F}$, 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.