= Affine invariance of Hotelling's statistic
{title2=$T_Y^2=T_X^2$}
Under a nonsingular affine transformation $Y=AX+b$, transform the null mean by the same map. Then the centered sample mean becomes $A(\bar X-\mu_0)$ and the <sample covariance matrix> becomes $ASA^T$. Since $(ASA^T)^{-1}=A^{-T}S^{-1}A^{-1}$, the quadratic form in <Hotelling's T-squared statistic> is unchanged. Its value and test decision are consequently invariant under changes of units, origin and nonsingular coordinates.
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