In my research, I encounter the following formula which I believe is correct (checked for $n\le3$). Is it classical ? If so, what is a reference ?

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 $\sigma_n(S)$ the determinant. The following formula gives $\sigma_k(S)$ in terms of the Gram matrix $G_k(s_1,\ldots,s_k)$ whose entries are the scalar products $Y(s_i)\cdot Y(s_j)$.

$$\sigma_k(S)=\frac1{k!}\int^{\otimes k}\det G_k(s_1,\ldots,s_k)\,d\mu(s_1)\cdots d\mu(s_k).$$

Remark that $S$ is positive semi-definite. The integrand is non-negative, as well as $\sigma_k(S)$. The integrand vanishes identically iff $Y(t)$ takes values in a subspace of dimension $<k$, which is the condition under which $\sigma_k(S)$ vanishes. It follows that, if the formula above failed, it would be because of an inequality between strictly positive numbers.

**Edit**. I delete the question mark in the formula above, because Marcel's answer and my comment to it, yield a proof.

ifthe proposed equation is false, then the it failsat mostin the form (positive real number)$\neq$(some other positive real number)'? I.e., it is for sure that it cannot fail in the form $0\neq$(positive real number)? $\endgroup$