Let \(A,B\) be \(m \times n\) matrices over the field of complex numbers and let \(C=A+B\). A number of subadditive inequalities between the eigenvalues/singular values of \(A,B\) and \(C\) have been discovered by many mathematicians, that include K. Fan, V. B. Lidskii, H. Wielandt, R. C. Thompson, A. Horn, over a period of time. In Section 2 of the paper, the authors review some of these inequalities including some majorization inequalities. In Section 4 the authors prove some analogous inequalities for canonical angles between Grassmannian subspaces. The bridge between singular values and the canonical angles is given by a ``multiplicative Pythagorean identity'' relating the direct rotations between three subspaces.