I see a similar formula I wish to be true and merely have a proof in mind occur as a MO’s problem: In my research, I encounter the following formula which I believe is correct (checked for $ n\le3$). Is it classical ? I am given a real symmetric matrix $ S:=\int Y(t)Y(t)^Td\mu(t),$ where $ \mu$ is a probability and $ Y(t):\Omega\rightarrow{\mathbb R}^n$. Let $ \sigma_k(S)$ be the elementary symmetric polynomial in the eigenvalues of $ S$. For instance, $ \sigma_1(S)$ is the trace and $ \si…
