# How to prove that Normal Squared Distances follow a Chi-Square distribution?

Given a multivariate normal distribution

$$f(x) = \frac{1}{\sqrt{(2 \pi)^n|\Sigma|}} \times \exp\left( -\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right)$$

how can I prove that $$(x-\mu)' \Sigma^{-1} (x-\mu) \sim \chi^2_{p}$$?

Note that a random vector $$X\sim N(\mu,\Sigma)$$ has the pdf in your post when $$\Sigma$$ is assumed to be positive definite. So there exists a nonsingular matrix $$P$$ such that $$\Sigma=PP'$$.
\begin{align} (X-\mu)' \Sigma^{-1}(X-\mu)&=(X-\mu)'(P')^{-1}P^{-1}(X-\mu) \\&=(X-\mu)'(P^{-1})'P^{-1}(X-\mu) \\&=Y'Y \end{align}
, where $$Y=P^{-1}(X-\mu)$$.
Now if you can show that $$Y$$ itself has a multivariate normal distribution with mean vector $$E(Y)=0$$ and dispersion matrix $$P^{-1}\Sigma (P^{-1})'=P^{-1}PP' (P')^{-1}=I$$, you are done.