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

Tomasz Kania
2015 The American mathematical monthly  
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