In this previous post, we defined projections onto a subspace and obtained an expression for the matrix representing this projection. Specifically, if is a full rank matrix and is the projection of onto the column space of , then , where . In this post, we generalize the idea of projections to Mahalanobis-type metrics.

Let be a symmetric positive definite matrix, and define the metric by for . Let be some full-rank matrix with . We can define the **projection onto the column space of using the metric **, denoted by , as the matrix such that for all , , where minimizes the quantity . That is, the projection of is the vector in the column space of that is closest to in the metric .

We can determine the projection matrix explicitly. Differentiating the objective function by and setting it to zero,

Since has full rank and is non-singular, is non-singular and so . This implies that

**How does this relate to our previous post?** Setting reduces to our initial expression for the projection matrix, . That is, our original expression for the projection matrix was for the (orthogonal) projection according to the standard Euclidean metric.

References:

- Ferguson, T. S. (1996).
*A Course in Large Sample Theory*. Chapter 23.

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