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arXiv · 1402.4272

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

Tomasz Kania
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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 $\mathbb{C}^n$.

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Tomasz Kania. 2014-02-18. A short proof of the fact that the matrix trace is the expectation of the numerical values. https://arxiv.org/abs/1402.4272

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