# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hamiltonian square roots of skew-Hamiltonian matrices revisited. (English) Zbl 0978.15010

Denote by ${M}_{n}\left(𝐂\right)$ and ${I}_{n}$ respectively the set of $n×n$ complex matrices and the identity matrix of order $n$. Set $J=\left(\begin{array}{cc}0& {I}_{n}\\ {I}_{n}& 0\end{array}\right)$. A matrix $H\in {M}_{2n}\left(𝐂\right)$ is Hamiltonian (resp. skew-Hamiltonian) if $H=\left(\begin{array}{cc}E& F\\ G& Y\end{array}\right)$ with blocks in ${M}_{n}\left(𝐂\right)$ where ${F}^{T}=F$, ${G}^{T}=G$, $Y=-{E}^{T}$ (resp. ${F}^{T}=-F$, ${G}^{T}=-G$, $Y={E}^{T}$); one has ${\left(JH\right)}^{T}=JH$ (resp. ${\left(JH\right)}^{T}=-JH$). A matrix $S\in {M}_{2n}\left(𝐂\right)$ is symplectic if ${S}^{T}JS=J$.

The author proves that every skew-Hamiltonian matrix can be brought into skew-Hamiltonian Jordan form $\left(\begin{array}{cc}K& 0\\ 0& {K}^{T}\end{array}\right)$ where $K\in {M}_{n}\left(𝐂\right)$ is in complex Jordan form, by a symplectic similarity transformation. This is the complex analog of a similar result of H. Fassbender, D. S. Mackey, N. Mackey, and 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 $Z\in {M}_{2n}\left(𝐂\right)$ is representable in the forms $WS$ and ${S}^{\text{'}}{W}^{\text{'}}$; here $S$, ${S}^{\text{'}}$ are symplectic and $W$, ${W}^{\text{'}}$ are skew-Hamiltonian.

##### MSC:
 15A24 Matrix equations and identities 15A21 Canonical forms, reductions, classification 15A57 Other types of matrices (MSC2000)