Some generalizations of the DDVV and BW inequalities

In this paper we generalize the known De Smet, Dillen, Verstraelen and Vrancken (DDVV)-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the Böttcher-Wenzel inequality to quaternionic matrices.