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A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values

2015
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The American mathematical monthly
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Using the fact that the normalised matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA) for all n × 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 Ax, x , where x ranges the unit sphere of C n .

doi:10.4169/amer.math.monthly.122.8.782
fatcat:ir4czrnuiffq7gw4v64oh25suu