  In this paper we generalize the known DDVV-type inequalities for real
(skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real,
complex and quaternionic matrices. Inspired by the Erd\H{o}s-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 B\"{o}ttcher-Wenzel inequality to quaternionic matrices.
