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Using the fact that the normalised matrix trace is the unique linearfunctional $f$ on the algebra of $n\times n$ matrices which satisfies $f(I)=1$and $f(AB)=f(BA)$ for all $n\times n$ matrices $A$ and $B$, we derive awell-known formula expressing the normalised trace of a complex matrix $A$ asthe expectation of the numerical values of $A$; that is the function $\langleAx,x\rangle$, where $x$ ranges the unit sphere of $\mathbb{C}^n$.

A short proof of the fact that the matrix trace is the expectation of the numerical values