Suppose that we have $n$ vectors $x_1,\ldots,x_n$ (each of dimension $p\times 1$). Let $$ \bar{x}\equiv\frac{1}{n}\sum_{i=1}^nx_i,\quad A\equiv\sum_{i=1}^n(x_i-\bar{x})(x_i-\bar{x})',\quad\underset{p\times 1}{\bar{x}^*}\equiv\begin{pmatrix}\sqrt{\bar{x}'A^{-1}\bar{x}} \\ 0 \\ \vdots \\ 0\end{pmatrix}. $$
Why is there a nonsingular $p\times p$ matrix $C$ such that both (i) and (ii) below are true? $$ \text{(i)}:\quad C\bar{x}=\bar{x}^*;\quad\quad\quad\text{(ii)}:\quad CAC'=I_p. $$ (This appears in passing in Anderson's Intro to Multivariate Analysis (page 191, 3rd edition.)
I can show such an orthogonal $C$ exists such that (i) is true but cannot see how that $C$ satisfies (ii). Similarly, I can show that some nonsingular $C$ exists such that (ii) is true but cannot see how to evaluate $C\bar{x}$.
Since $A$ is positive matrix you can have $B$ positive matrix $B^2 = A$ and $B= B'$. $$\left\|B^{-1}\cdot \bar x\right\|^2 = \langle B^{-1}\cdot \bar x, B^{-1}\cdot \bar x\rangle = \langle \bar x, B^{-2} \bar x\rangle = \bar x'A^{-1}\bar x = \left\|\bar x^*\right\|^2$$ then you can have an unitary matrix $U$ such that $$U\left(B^{-1}\cdot \bar x\right) = \bar x^*.$$ Take $C=UB^{-1}$. $$CAC' = UB^{-1}AB^{-1}U' = UU' = I_p$$