In this post is the standard inner product on for some and is a fixed Hermitian positive definite operator. The square Mahalanobis norm of a vector for this operator is defined by
The results in this post were found while looking for ways to approximate this Mahalanobis norm without the need to invert . (Later I realised that using the Cholesky factorisation suited me better. Nice results can be found by looking in the wrong places!) The idea is to use projections on some smaller dimensional subspace to get estimates of the actual Mahalanobis norm. To be precise let be some subspace of dimension and let be the orthogonal projection onto . The operator is non-singular on the subspace . Let be its pseudo inverse such that . The projected Mahalanobis norm on is defined by
Let’s take the one-dimensional case as an example. Let be non-zero and denote the span of by . Then the norm is given by
Note that this expression does not involve the inverse of . The basic property of the projected Mahalanobis norm is the following:
The inequality holds throughout . Equality occurs if and only if .
This property follows from the Cauchy-Schwarz inequality for the inner product :
This is an equality if and only if and are linearly dependent. Combined with it follows that in fact .
The following realisation came as a surprise. It shows that projections onto two-dimensional subspaces suffice to get an exact value for the Mahalanobis norm:
Let be a non-zero vector and let be the span of (so ). Then .
The projected norm for a two-dimensional subspace also has a simple explicit form. Let be a non-zero vector orthogonal to and let be the span of . The norm is given by