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