I know quite a few identities about quadratic forms of random vectors, but I'm having difficulty coaxing something out of this quadratic form of random matrices. Suppose I know $\mathbb{E}[W W^{T}]$ and $\mathbb{E}[W] = 0$. Then can I deduce a closed for form $\mathbb{E}[W^{T} P W]$ for a non-random matrix $P$?
EDIT Changed given from $\mathbb{E}[W^{T} W]$ to $\mathbb{E}[W W^{T}]$, added $\mathbb{E}[W] = 0$.