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| | Friends call him Royal. Delaware has usually been my living location and will by no means move. She works as a monetary officer and she will not change it whenever quickly. Playing crochet is some thing that I've carried out for many years.<br><br>Feel free to surf to my web page: [http://changbai.Newdrycleaning.com/photo/54180 changbai.Newdrycleaning.com] |
| In [[mathematics]], a '''Hamiltonian matrix''' is a {{math|2''n''}}-by-{{math|2''n''}} [[matrix (mathematics)|matrix]] {{math|''A''}} such that {{math|''JA''}} is [[symmetric matrix|symmetric]], where {{math|''J''}} is the [[skew-symmetric]] matrix
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| :<math>J=
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| \begin{bmatrix}
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| 0 & I_n \\
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| -I_n & 0 \\
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| \end{bmatrix}</math>
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| and {{math|''I<sub>n</sub>''}} is the {{math|''n''}}-by-{{math|''n''}} [[identity matrix]]. In other words, {{math|''A''}} is Hamiltonian if and only if {{math|1 = (''JA'')<sup>T</sup> = ''JA''}} where {{math|()<sup>T</sup>}} denotes the [[transpose]].<ref name=ikramov>{{citation | first = Khakim D. | last = Ikramov | title = Hamiltonian square roots of skew-Hamiltonian matrices revisited | journal = Linear Algebra and its Applications | volume = 325 | year = 2001 | pages = 101–107 | doi = 10.1016/S0024-3795(00)00304-9 }}.</ref>
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| ==Properties==
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| Suppose that the {{math|2''n''}}-by-{{math|2''n''}} matrix {{math|''A''}} is written as the [[block matrix]]
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| :<math> A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}</math>
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| where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, and {{math|''d''}} are {{math|''n''}}-by-{{math|''n''}} matrices. Then the condition that {{math|''A''}} be Hamiltonian is equivalent to requiring that the matrices {{math|''b''}} and {{math|''c''}} are symmetric, and that {{math|1 = ''a'' + ''d''<sup>T</sup> = 0}}.<ref name=ikramov /><ref name=meyer>{{citation | first1 = K. R. | last1 = Meyer | first2 = G. R. | last2 = Hall | title = Introduction to Hamiltonian dynamical systems and the {{math|''N''}}-body problem | publisher = [[Springer Science+Business Media|Springer]] | year = 1991 | isbn = 0-387-97637-X }}.</ref> Another equivalent condition is that {{math|''A''}} is of the form {{math|1 = ''A'' = ''JS''}} with {{math|''S''}} symmetric.<ref name=meyer />{{rp|34}}
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| It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any [[linear combination]]) of two Hamiltonian matrices is again Hamiltonian, as is their [[commutator]]. It follows that the space of all Hamiltonian matrices is a [[Lie algebra]], denoted {{math|sp(2''n'')}}. The dimension of {{math|sp(2''n'')}} is {{math|2''n''<sup>2</sup> + ''n''}}. The corresponding [[Lie group]] is the [[symplectic group]] {{math|Sp(2''n'')}}. This group consists of the [[symplectic matrix|symplectic matrices]], those matrices {{math|''A''}} which satisfy {{math|1 = ''A''<sup>T</sup>''JA'' = ''J''}}. Thus, the [[matrix exponential]] of a Hamiltonian matrix is symplectic, and the logarithm of a symplectic matrix is Hamiltonian.<ref name=meyer />{{rp|34–36}}<ref name=dragt>{{citation | first = Alex J. | last = Dragt | doi = 10.1196/annals.1350.025 | title = The symplectic group and classical mechanics | journal = Annals of the New York Academy of Sciences | year = 2005 | volume = 1045 | issue = 1 | pages=291–307}}.</ref>
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| The [[characteristic polynomial]] of a real Hamiltonian matrix is [[even function|even]]. Thus, if a Hamiltonian matrix has {{math|λ}} as an [[eigenvalue]], then {{math|−λ}}, {{math|λ<sup>*</sup>}} and {{math|−λ<sup>*</sup>}} are also eigenvalues.<ref name=meyer />{{rp|45}} It follows that the [[trace (linear algebra)|trace]] of a Hamiltonian matrix is zero.
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| The square of a Hamiltonian matrix is [[skew-Hamiltonian matrix|skew-Hamiltonian]] (a matrix {{math|''A''}} is skew-Hamiltonian if {{math|1 = (''JA'')<sup>T</sup> = −''JA''}}). Conversely, every skew-Hamiltonian matrix arises as the square of a Hamiltonian matrix.<ref name=waterhouse>{{citation | first = William C. | last = Waterhouse | authorlink = William C. Waterhouse | doi = 10.1016/j.laa.2004.10.003 | title = The structure of alternating-Hamiltonian matrices | journal = Linear Algebra and its Applications | volume = 396 | year = 2005 | pages = 385–390 }}.</ref>
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| ==Extension to complex matrices==
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| The definition for Hamiltonian matrices can be extended to complex matrices in two ways. One possibility is to say that a matrix {{math|''A''}} is Hamiltonian if {{math|1 = (''JA'')<sup>T</sup> = ''JA''}}, as above.<ref name=ikramov /><ref name=waterhouse /> Another possibility is to use the condition {{math|1 = (''JA'')<sup>*</sup> = ''JA''}} where {{math|()<sup>*</sup>}} denotes the [[conjugate transpose]].<ref name=paige>{{citation | first1 = Chris | last1 = Paige | first2 = Charles | last2 = Van Loan | authorlink2 = Charles F. Van Loan | title = A Schur decomposition for Hamiltonian matrices | journal = Linear Algebra and its Applications | volume = 41 | year = 1981 | pages = 11–32 | doi = 10.1016/0024-3795(81)90086-0 }}.</ref>
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| ==Hamiltonian operators==
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| Let {{math|''V''}} be a vector space, equipped with a symplectic form {{math|Ω}}. A linear map <math>A:\; V \mapsto V</math> is called '''a Hamiltonian operator''' with respect to {{math|Ω}} if the form <math>x, y \mapsto \Omega(A(x), y)</math> is symmetric. Equivalently, it
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| should satisfy
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| :<math>\Omega(A(x), y)=-\Omega(x, A(y))</math>
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| Choose a basis {{math|''e''<sub>1</sub>, …, ''e''<sub>2''n''</sub>}} in {{math|''V''}}, such that {{math|Ω}} is written as <math>\sum_i e_i \wedge e_{n+i}</math>. A linear operator is Hamiltonian with respect to {{math|Ω}} if and only if its matrix in this basis is Hamiltonian.<ref name=waterhouse />
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| ==References==
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| <references />
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| [[Category:Matrices]]
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Friends call him Royal. Delaware has usually been my living location and will by no means move. She works as a monetary officer and she will not change it whenever quickly. Playing crochet is some thing that I've carried out for many years.
Feel free to surf to my web page: changbai.Newdrycleaning.com