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• 1402.4272v1

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

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• Tomasz Kania
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Journal publication date 10/2015 American Mathematical Monthly 8 122 2 782-783 Published 24/02/14 English

### Abstract

Using the fact that the normalised matrix trace is the unique linear functional $f$ 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 a well-known formula expressing the normalised trace of a complex matrix $A$ as the expectation of the numerical values of $A$; that is the function $\langle Ax,x\rangle$, where $x$ ranges the unit sphere of $\mathbb{C}^n$.