= Solution
Assume an independent Gaussian sample, positive-definite population covariance, and $n>p$. Define the <sample mean> and unbiased <sample covariance matrix> by
$$
\bar X=\frac1n\sum_{i=1}^nX_i,\qquad S=\frac1{n-1}\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X)^T.
$$
The standard test is based on <Hotelling's T-squared statistic>
$$
\boxed{T^2=n(\bar X-\mu_0)^TS^{-1}(\bar X-\mu_0).}
$$
Reject the <null hypothesis> for large values. Under the null its exact distribution is
$$
\boxed{\frac{n-p}{p(n-1)}T^2\sim F_{p,n-p}.}
$$
The factor uses the unbiased, divisor-$n-1$ covariance convention. Normal-sample theory makes $\bar X$ independent of the residual covariance, whose whitened version has a <Wishart distribution>; this yields the stated <F-distribution>. The condition $n>p$ gives invertibility of $S$ almost surely and positive denominator degrees of freedom.
This is also the <likelihood-ratio test>. To see the connection, put $W=(n-1)S$ and $\delta=\bar X-\mu_0$. The maximized covariance estimates under the unrestricted and null models are $W/n$ and $W/n+\delta\delta^T$, respectively. The determinant identity for a rank-one update gives
$$
\frac{|W/n+\delta\delta^T|}{|W/n|}=1+n\delta^TW^{-1}\delta=1+\frac{T^2}{n-1}.
$$
Thus the likelihood ratio is $(1+T^2/(n-1))^{-n/2}$, decreasing in $T^2$. The term best is understood here as this standard likelihood-ratio, affine-invariant procedure; a two-sided multidimensional alternative does not specify a single direction to optimize power against.
For the transformed sample $Y_i=AX_i+b$, with $A$ the prescribed nonsingular coordinate transformation, we have
$$
\bar Y-(A\mu_0+b)=A(\bar X-\mu_0),\qquad S_Y=ASA^T,\qquad S_Y^{-1}=A^{-T}S^{-1}A^{-1}.
$$
Consequently
$$
T_Y^2=n\delta^TA^T(A^{-T}S^{-1}A^{-1})A\delta=n\delta^TS^{-1}\delta=T_X^2.
$$
This proves the <affine invariance of Hotelling's statistic>. \b[The statistic, its p-value and its test decision are unchanged by a nonsingular affine transformation], provided the null mean is transformed with the data. Changes of units and rotations are included.
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