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Zbl 0978.15010
Ikramov, Khakim D.
Hamiltonian square roots of skew-Hamiltonian matrices revisited.
(English)
[J] Linear Algebra Appl. 325, No. 1-3, 101-107 (2001). ISSN 0024-3795

Denote by $M_n({\bold C})$ and $I_n$ respectively the set of $n\times n$ complex matrices and the identity matrix of order $n$. Set $J=\left(\smallmatrix 0&I_n\\ I_n&0\endsmallmatrix\right)$. A matrix $H\in M_{2n}({\bold C})$ is Hamiltonian (resp. skew-Hamiltonian) if $H=\left(\smallmatrix E&F\\ G&Y \endsmallmatrix\right)$ with blocks in $M_n({\bold C})$ where $F^T=F$, $G^T=G$, $Y=-E^T$ (resp. $F^T=-F$, $G^T=-G$, $Y=E^T$); one has $(JH)^T=JH$ (resp. $(JH)^T=-JH$). A matrix $S\in M_{2n}({\bold C})$ is symplectic if $S^TJS=J$.\par The author proves that every skew-Hamiltonian matrix can be brought into skew-Hamiltonian Jordan form $\left(\smallmatrix K&0\\0&K^T\endsmallmatrix\right)$ where $K\in M_n({\bold C})$ is in complex Jordan form, by a symplectic similarity transformation. This is the complex analog of a similar result of {\it H. Fassbender, D. S. Mackey, N. Mackey}, and {\it H. Xu} in the case of real matrices [Linear Algebra Appl. 287, No. 1-3, 125-159 (1999; Zbl 0940.15017)]. The author proves also that every skew-Hamiltonian matrix has a Hamiltonian square root, that two similar skew-Hamiltonian matrices $U$, $V$ are symplectic similar, i.e. $U=S^{-1}VS$, and that every nonsingular matrix $Z\in M_{2n}({\bold C})$ is representable in the forms $WS$ and $S'W'$; here $S$, $S'$ are symplectic and $W$, $W'$ are skew-Hamiltonian.
[V.P.Kostov (Nice)]
MSC 2000:
*15A24 Matrix equations
15A21 Canonical forms, etc.
15A57 Other types of matrices

Keywords: Hamiltonian matrices; skew-Hamiltonian matrices; symplectic matrices; Jordan form; symplectic similarity; Hamiltonian square root

Citations: Zbl 0940.15017

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