Uniqueness of matrix square roots and an application

Let A is an element of M-n (C). Let sigma (A) denote the spectrum of A, and F(A) the field of values of A. It is shown that if sigma (A) similar to (- infinity ,0] = empty set, then A has a unique square root B is an element o f M-n (C) with sigma (B) in the open right (complex) half plane. This resul t and Lyapunov's theorem are then applied to prove that if F(A) boolean AND (-infinity, 0] = empty set, then A has a unique square root with positive definite Hermitian part. We will also answer affirmatively an open question about the existence of a real square root B is an element of M-n (R) for A is an element of M-n (R) with F(A) n (-infinity ,0] = empty set where the field of values of B is in the open right half plane. (C) 2001 Elsevier Sci ence Inc. All rights reserved, AMS classification: 15A18; 15A21; 15A24; 15A 57.