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Zbl 0976.15009
Johnson, Charles R.; Okubo, Kazuyoshi; Reams, Robert
Uniqueness of matrix square roots and an application.
(English)
[J] Linear Algebra Appl. 323, No.1-3, 51-60 (2001). ISSN 0024-3795

The question of uniqueness of square roots is studied for matrices $A \in M_n( {\bbfC})$ with the set of eigenvalues $\sigma(A)$ and the field of values defined as $F(A)=[x^*Ax: x^*x=1, x \in {\bbfC} ^n]$. There are given simplified proofs that if $\sigma(A)$ is a part of the open right half of the complex plane (or more generally, $\sigma(A) \cap (- \infty, 0] = \emptyset$) then there is a square root of $A$, $A^{1/2}$, such that $\sigma(A^{1/2})$ lies in the open right half of the complex plane. It is also shown, using Lyapunov's theorem, that if $A \in M_n( {\bbfC})$ and the Hermitian part of $A$, $H(A) = \frac {1}{2}(A + A^*)$, is positive definite (or more generally, $F(A) \cap (-\infty,0] = \emptyset$) then there is a square root of $A$, $A^{1/2}$, such that its Hermitian part $H(A^{1/2})$ is positive definite. \par The open question mentioned by {\it C. R. Johnson} and {\it M. Neumann} [Linear Multilinear Algebra 8, 353-355 (1980; Zbl 0431.15011)] whether $A \in M_n({\bbfR})$, such that $H(A)$ is positive definite, has a square root $A^{1/2} \in M_n({\bbfR})$, such that $H(A^{1/2})$ is positive definite, is answered affirmatively.
[Václav Burjan (Praha)]
MSC 2000:
*15A24 Matrix equations

Keywords: matrix square root; Jordan canonical form; Lyapunov's theorem; field of values

Citations: Zbl 0431.15011

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