On triangle inequality for Miranda-Thompson’s majorization and gradients of increasing functions

Abstract

In this paper, we establish some refinements of the following triangle inequality proved by T.-Y. Tam: $$ {{\widetilde{\sigma }}} (x + y) \prec _{mt} {{\widetilde{\sigma }}} (x) + {{\widetilde{\sigma }}} (y) $$ for real matrices x and y with the modified singular value operator $$ {{\widetilde{\sigma }}} (\cdot ) $$ and Miranda-Thompson’s majorization $$ \prec _{mt} $$ on $$ {\mathbb {R}}^{n}$$ related to the group of special orthogonal matrices.