# Show that distribution of $\small(n-1)\overline{X}'(S^{-1}-\frac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0})\overline{X}$ is $\small T^2(p-1,n-1)$

Show that the distribution of $$(n-1)\overline{X}'\left(S^{-1}-\dfrac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0}\right)\overline{X}$$ is $$T^2(p-1,n-1)$$ where $$X_i\sim N_p(\mu,\Sigma)$$, where $$\Sigma$$ is pd but unknown, and it is just known that $$\mu=k\mu_0$$ for some $$k\in\mathbb R$$. Also, $$nS=\sum_{i=1}^n(X_i-\overline{X})(X_i-\overline{X})'$$.

I know a solution where I can view $$\mu=k\mu_0$$ as $$R\mu=0$$ for some suitable matrix $$R$$ orthogonal to $$\mu_0$$, but I do not want to use that result. I want to directly obtain the result using properties of Wishart and its relation with Hotelling $$T^2$$, etc.

I know that $$nS\sim W_p(\Sigma,n-1)$$ and is independent of $$\overline{X}$$. I will be done if I can somehow identify the middle matrix as inverse of some Wishart with appropriate parameters.

Can anyone give some hint? Thanks.