On a subspace metric based on matrix rank

The metric between subspaces M,N⊆Cn,1, defined by δ(M,N)=rk(PM-PN), where rk(·) denotes rank of a matrix argument and PM and PN are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ(M,N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.