I am studying the textbook An Introduction to Mutivariate Statistical analysis by T.W Anderson and was stuck by the following exercise
$\textbf{7.13}$ Let $Z_1,\ \dots,\ Z_n$ be independently distributed, each according to $N(0,\ I)$. Let $W=\sum\limits_{\alpha=1}^n \sum\limits_{\beta=1}^n b_{\alpha\beta}Z_{\alpha}Z_{\beta}^{T}$. Prove that if $a^TWa \sim \chi_m^2$ for all $a$ such that $a^Ta=1$, then $W$ is distributed according to $W(I,\ m)$.
Here is my attempt: We denote by $Z=(Z_1,\ \dots, \ Z_n)^T$ and $B=(b_{\alpha\beta})_{\alpha,\ \beta=1}^n$, then $W=Z^TBZ$, $a^TWa=(Za)^TB(Za)$ and $Za\sim N(0,\ I_n)$ for all $a$ with $a^Ta=1$. Let $C=\dfrac{B+B^T}{2}$, then $C$ is symmetric and $(Za)^TB(Za)=(Za)^TC(Za)$. It is easy to show that $C$ is idempotent with rank $m$. However, $Z^TBZ$ does not necessarily euqal $Z^TCZ$ and I did not know how to proceed.
Can anyone help me?
