I want to show that $cov(\frac{1}{2}\textbf{Z}'A\textbf{Z},\frac{1}{2}\textbf{Z}'B\textbf{Z})=\frac{1}{2}tr(AR(\theta)BR(\theta))$

where $A,B \in Sym(p)$ (real symmetric $p \times p$ matrices) and $\textbf{Z}= (Z_1 , \ldots,Z_p)'\sim N_p(0,R(\theta))$

I tried to use the fact that if $x \sim N_n (\mu, \Omega)$ and $A$ is a symmetric $n \times n$ matrix, then $$ \text{E}[x'Ax] = tr A \Omega + \mu ' A \mu.$$

but i didn't succeed. Do you have any suggestions?


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