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.