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Hamiltonian square roots of skew-Hamiltonian matrices revisited. (English) Zbl 0978.15010
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 $S^TJS=J$. 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.

15A24Matrix equations and identities
15A21Canonical forms, reductions, classification
15B57Hermitian, skew-Hermitian, and related matrices
Full Text: DOI
[1] Fassbender, H.; Mackey, D. S.; Mackey, N.; Xu, H.: Hamiltonian square roots of skew-Hamiltonian matrices. Linear algebra appl. 287, 125-159 (1999) · Zbl 0940.15017
[2] H. Fassbender, D.S. Mackey, N. Mackey, H. Xu, Real and complex Hamiltonian square roots of skew-Hamiltonian matrices, Mathematics and Statistics Reports, #92, Western Michigan University, 1999 · Zbl 0940.15017
[3] Gantmacher, F. R.: The theory of matrices. (1959) · Zbl 0085.01001
[4] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1990) · Zbl 0704.15002
[5] Horn, R. A.; Johnson, C. R.: Topics in matrix analysis. (1991) · Zbl 0729.15001
[6] Ikramov, H. D.: Linear algebra. Problems book. (1983) · Zbl 0523.15002