Proof of the rank-trace theorem

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A and use the Cauchy-Schwarz inequality. Detailed proof Now let’s