ITERATED EXPONENTIATION, MATRIX-MATRIX EXPONENTIATION, AND ENTROPY

[摘要] On [0, infinity), log(x(xxx)) is strictly convex. Matrix-matrix exponentiation A(B) 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(A(A))]. Using convexity, an obvious generalization of S(A), namely -log[det(A(AAA)], 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. (C) 1994 Academic Press, Inc.

[发布日期] 1994-04-01