On the angular metrics between linear subspaces

It has recently been shown that the symmetric gauge functions on the canonical (principal) angles give a family of unitarily invariant metrics between linear subspaces of the same dimension. In this short paper, we extend such metrics to subspaces of possibly different dimensions. This extension is necessary in addressing some perturbation analysis problems involving subspaces with different dimensions. Examples of such perturbation analysis problems are also studied in this paper using the extended metrics.