Uniqueness of matrix square roots and an application

By Charles R. Johnson, K. Okubo, Robert Reams. 2001

58 citations recorded by the source corpus.

Abstract Let A∈M n ( C ) . 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]=∅ , then A has a unique square root B∈M n ( C ) with σ(B) in the open right (complex) half plane. This result and Lyapunov's theorem are then applied to prove that if F(A)∩(−∞,0]=∅ , 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∈M n ( R ) for A∈M n ( R ) with F(A)∩(−∞,0]=∅ , where the field of values of B is in the open right half plane.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/78134728457d7640d3e3ba6119a980de78863e1c