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modifying the Mahalanobis distance

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I know that for a Gaussian distribution with covariance matrix K and mean m, the mahalanobis distance from a point to this Gaussian is computed as (y-m)TK-1(y-m). What if I also want to scale the distance by another symmetric matrix C, would this be a valid way of modifying the Mahalanobis distance: (y-m)TCK-1C(y-m) or (y-m)TC1/2K-1C1/2(y-m) where C = VDVT (eigen decomposition) and C1/2 = VD1/2VT. If the matric C only has value on the diagonal, this simply reduces to scaling the parameters of y, which is where I got the idea from. Any thoughts would be appreciated. Shaobo

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