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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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<mark>Journal publication date</mark>10/2015
<mark>Journal</mark>American Mathematical Monthly
Issue number8
Volume122
Number of pages2
Pages (from-to)782-783
Publication StatusPublished
Early online date24/02/14
<mark>Original language</mark>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 $\mathbb{C}^n$.