Uniqueness of matrix square roots and an application Original Research Article

Let image. Let σ(A) denote the spectrum of A, and F(A) the field of values of A. It is shown that if σ(A)∩(−∞,0]=empty set︀, then A has a unique square root image with σ(B) in the open right (complex) half plane. This result and Lyapunovʹs theorem are then applied to prove that if F(A)∩(−∞,0]=empty set︀, 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 image for image with F(A)∩(−∞,0]=empty set︀, where the field of values of B is in the open right half plane.