On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart–Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart–Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.