From Subadditive Inequalities of Singular Values to Triangle Inequalities of Canonical Angles

The singular values of matrices $A,B,C\in\mathbb{C}^{m\times n}$ with $C=A+B$ satisfy an extensive list of subadditive inequalities discovered by K. Fan, V.B. Lidskii, H. Wielandt, R.C. Thompson, A. Horn, and so on. These inequalities still hold when we apply a nonnegative concave function to each of the singular values involved, as shown recently by M. Uchiyama and J.C. Bourin. The main purpose of this paper is to show that all of these singular value inequalities can be translated into canonical angle inequalities. The bridge between the singular values and the canonical angles is given by a “multiplicative Pythagorean identity” relating the direct rotations between three subspaces.