A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values
Tomasz Kania · American Mathematical Monthly · 2015
Using the fact that the normalized 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 normalized 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 ℂn.