*(English)*Zbl 0976.15009

The question of uniqueness of square roots is studied for matrices $A\in {M}_{n}\left(\u2102\right)$ with the set of eigenvalues $\sigma \left(A\right)$ and the field of values defined as $F\left(A\right)=[{x}^{*}Ax:{x}^{*}x=1,x\in {\u2102}^{n}]$. There are given simplified proofs that if $\sigma \left(A\right)$ is a part of the open right half of the complex plane (or more generally, $\sigma \left(A\right)\cap (-\infty ,0]=\varnothing $) then there is a square root of $A$, ${A}^{1/2}$, such that $\sigma \left({A}^{1/2}\right)$ lies in the open right half of the complex plane. It is also shown, using Lyapunov’s theorem, that if $A\in {M}_{n}\left(\u2102\right)$ and the Hermitian part of $A$, $H\left(A\right)=\frac{1}{2}(A+{A}^{*})$, is positive definite (or more generally, $F\left(A\right)\cap (-\infty ,0]=\varnothing $) then there is a square root of $A$, ${A}^{1/2}$, such that its Hermitian part $H\left({A}^{1/2}\right)$ 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 $A\in {M}_{n}\left(\mathbb{R}\right)$, such that $H\left(A\right)$ is positive definite, has a square root ${A}^{1/2}\in {M}_{n}\left(\mathbb{R}\right)$, such that $H\left({A}^{1/2}\right)$ is positive definite, is answered affirmatively.

##### MSC:

15A24 | Matrix equations and identities |