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Uniqueness of matrix square roots and an application. (English) Zbl 0976.15009

The question of uniqueness of square roots is studied for matrices AM n () with the set of eigenvalues σ(A) and the field of values defined as F(A)=[x * Ax:x * x=1,x n ]. There are given simplified proofs that if σ(A) is a part of the open right half of the complex plane (or more generally, σ(A)(-,0]=) then there is a square root of A, A 1/2 , such that σ(A 1/2 ) lies in the open right half of the complex plane. It is also shown, using Lyapunov’s theorem, that if AM n () and the Hermitian part of A, H(A)=1 2(A+A * ), is positive definite (or more generally, F(A)(-,0]=) then there is a square root of A, A 1/2 , such that its Hermitian part H(A 1/2 ) is positive definite.

The open question mentioned by C. R. Johnson and M. Neumann [Linear Multilinear Algebra 8, 353-355 (1980; Zbl 0431.15011)] whether AM n (), such that H(A) is positive definite, has a square root A 1/2 M n (), such that H(A 1/2 ) is positive definite, is answered affirmatively.

MSC:
15A24Matrix equations and identities