Affine invariance of Hotelling's statistic ($T_Y^2=T_X^2$)

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 $AS{A}^T$. Since $(AS{A}^T{)}^{-1}=A^{-T}{S}^{-1}{A}^{-1}$, 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.