I'm reading the papaer "On the Reduction of a Matrix to Diagonal Form" of Epstein and Flanders (Amer. Math. Monthly 62, (1955). 168–171.
Let $S$ denote the trace function.
The authors stated that a well-known result in the theory of algebras of matrices is:
A matrix algebra $\mathbb U$ over a field $\mathbb F$ of characteristice zero is semisimples if and only if $S(XY)=0$, for fixed $X\in\mathbb U$ and all $Y\in\mathbb U$,implies $X=0$.
The authors do not give references. Does anyone know where I can find a proof of this result?