Iterated Exponentiation, Matrix-Matrix Exponentiation, and Entropy

On [0, ∞), log(xxxx) is strictly convex. Matrix-matrix exponentiation AB is defined when A is normal and nonsingular. Von Neumann′s quantum-mechanical entropy S(A) of a density-matrix A can be written S(A) = −log[det(AA)]. Using convexity, an obvious generalization of S(A), namely −log[det(AAAA)], is shown to satisfy the same monotonicity inequality as S(A). Matrix-matrix exponentiation is used to generalize several results about iterated exponentiation of scalars.