We derive new characterizations for the matrix Φ-entropies introduced in [Electron. J. Probab., 19(20): 1–30, 2014]. The fact that these new characterizations are a direct generalization of their corresponding equivalent statements for classical Φ-entropies provides additional justification to the original definition of matrix Φ-entropies. Moreover, these extra characterizations allow us to better understand the properties of matrix Φ-entropies, which are a powerful tool for unifying the study matrix concentration inequalities. We then move on to prove a Poincare inequality for these matrix Φ-entropies.
Along the way, we also provide a new proof for the matrix Efron-Stein inequality. Finally, we derive a restricted logarithmic Sobolev inequality for matrix-valued functions defined on Boolean hypercubes. Our proof relies on the powerful matrix Bonami-Beckner inequality.
