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

作者：Kania Tomasz^*

来源：American Mathematical Monthly, 2015, 122(8): 782-+.

DOI：10.4169/amer.math.monthly.122.8.782

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

• 出版日期2015-10