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| In mathematics, a '''symplectic matrix''' is a 2''n''×2''n'' [[matrix (mathematics)|matrix]] ''M'' with [[real number|real]] entries that satisfies the condition
| | Msvcr71.dll is an important file that assists help Windows process different components of the program including important files. Specifically, the file is selected to help run corresponding files in the "Virtual C Runtime Library". These files are significant in accessing any settings which support the different applications plus programs in the system. The msvcr71.dll file fulfills numerous important functions; nevertheless it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a hard time processing plus reading components of the system. Nonetheless, consumers want not panic because this issue can be solved by following several procedures. And I can show you several tricks about Msvcr71.dll.<br><br>You may discover that there are registry products that are free and those that you'll have to pay a nominal sum for. Some registry products provide a bare bones program for free with the choice of upgrading to a more advanced, efficient adaptation of the same program.<br><br>So, this advanced double scan is not merely 1 of the greater, nevertheless it is also freeware. And as of all of this that many regard CCleaner one of the greater registry cleaners in the marketplace now. I would add that I personally like Regcure for the easy reason that it has a greater interface and I learn for a fact that it is ad-ware without charge.<br><br>Fixing tcpip.sys blue screen is easy to do with registry repair software.Trying to fix windows blue screen error on a own may be challenging considering should you remove or damage the registry it may cause serious damage to your computer. The registry should be cleaned and all erroneous and incomplete info removed to stop blue screen errors from occurring.The benefit of registry repair software is not limited to simply getting rid of the blue screen on business.We may be amazed at the greater and more improved speed and performance of the computer system following registry cleaning is done. Registry cleaning will really develop a computer's functioning abilities, especially when you choose a certain registry repair software that is especially efficient.<br><br>Another common cause of PC slow down is a corrupt registry. The registry is a pretty important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not commence at all. Fixing the registry is easy with all the employ of the program and [http://bestregistrycleanerfix.com/registry-mechanic registry mechanic].<br><br>Let's start with all the negative sides first. The initial price of the product is fairly inexpensive. However, it only comes with one year of updates. After which you need to subscribe to monthly updates. The advantage of that is the fact that ideal optimizer has enough funds plus resources to analysis mistakes. This way, we are ensured of safe fixes.<br><br>Reboot PC - Simply reboot a PC to see if the error is gone. Frequently, rebooting the PC readjusts the internal settings plus software plus therefore fixes the problem. If it doesn't then move on to follow the instructions under.<br><br>You are able to click here to locate out how to speed up Windows and strengthen PC perfomance. And you can click here to download a registry cleaner to aid you clean up registry. |
| {{NumBlk|:|<math>M^T \Omega M = \Omega\,.</math>|{{EquationRef|1}}}}
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| where ''M<sup>T</sup>'' denotes the [[transpose]] of ''M'' and Ω is a fixed 2''n''×2''n'' [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]]. This definition can be extended to 2''n''×2''n'' matrices with entries in other [[field (mathematics)|field]]s, e.g. the [[complex number]]s.
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| Typically Ω is chosen to be the [[block matrix]]
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| :<math>\Omega =
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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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| where ''I''<sub>n</sub> is the ''n''×''n'' [[identity matrix]]. The matrix Ω has [[determinant]] +1 and has an inverse given by Ω<sup>−1</sup> = Ω<sup>''T''</sup> = −Ω.
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| Every symplectic matrix has unit determinant, and the 2''n''×2''n'' symplectic matrices with real entries form a [[subgroup]] of the [[special linear group]] SL(2''n'', ''R'') under [[matrix multiplication]], specifically a [[connected space|connected]] [[compact space|noncompact]] [[real Lie group]] of real dimension {{nowrap|''n''(2''n'' + 1)}}, the [[symplectic group]] Sp(2''n'', '''R'''). The symplectic group can be defined as the set of [[linear transformations]] that preserve the symplectic form of a real [[symplectic vector space]].
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| ==Properties==
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| Every symplectic matrix is [[invertible matrix|invertible]] with the inverse matrix given by
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| :<math>M^{-1} = \Omega^{-1} M^T \Omega.</math>
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| Furthermore, the [[matrix multiplication|product]] of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a [[group (mathematics)|group]]. There exists a natural [[manifold]] structure on this group which makes it into a (real or complex) [[Lie group]] called the [[symplectic group]]. The symplectic group has dimension ''n''(2''n'' + 1).
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| It follows easily from the definition that the [[determinant]] of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1. One way to see this is through the use of the [[Pfaffian]] and the identity
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| :<math>\mbox{Pf}(M^T \Omega M) = \det(M)\mbox{Pf}(\Omega).</math>
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| Since <math>M^T \Omega M = \Omega</math> and <math>\mbox{Pf}(\Omega) \neq 0</math> we have that det(''M'') = 1.
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| Suppose Ω is given in the standard form and let ''M'' be a 2''n''×2''n'' [[block matrix]] given by
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| :<math>M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}</math>
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| where ''A, B, C, D'' are ''n''×''n'' matrices. The condition for ''M'' to be symplectic is equivalent to the conditions
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| :<math>A^TD - C^TB = I</math>
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| :<math>A^TC = C^TA</math>
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| :<math>D^TB = B^TD.</math>
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| When ''n'' = 1 these conditions reduce to the single condition det(''M'') = 1. Thus a 2×2 matrix is symplectic [[iff]] it has unit determinant.
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| With Ω in standard form, the inverse of ''M'' is given by
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| :<math>M^{-1} = \Omega^{-1} M^T \Omega=\begin{pmatrix}D^T & -B^T \\-C^T & A^T\end{pmatrix}.</math>
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| ==Symplectic transformations==
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| In the abstract formulation of [[linear algebra]], matrices are replaced with [[linear transformation]]s of [[finite-dimensional]] [[vector spaces]]. The abstract analog of a symplectic matrix is a '''symplectic transformation''' of a [[symplectic vector space]]. Briefly, a symplectic vector space is a 2''n''-dimensional vector space ''V'' equipped with a [[nondegenerate form|nondegenerate]], [[skew-symmetric]] [[bilinear form]] ω called the [[symplectic form]].
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| A symplectic transformation is then a linear transformation ''L'' : ''V'' → ''V'' which preserves ω, i.e.
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| :<math>\omega(Lu, Lv) = \omega(u, v).</math>
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| Fixing a [[basis (linear algebra)|basis]] for ''V'', ω can be written as a matrix Ω and ''L'' as a matrix ''M''. The condition that ''L'' be a symplectic transformation is precisely the condition that ''M'' be a symplectic matrix:
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| :<math>M^T \Omega M = \Omega.</math>
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| Under a [[change of basis]], represented by a matrix ''A'', we have
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| :<math>\Omega \mapsto A^T \Omega A</math>
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| :<math>M \mapsto A^{-1} M A.</math>
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| One can always bring Ω to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of ''A''.
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| ==The matrix Ω==
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| Symplectic matrices are defined relative to a fixed [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]] Ω. As explained in the previous section, Ω can be thought of as the coordinate representation of a [[nondegenerate form|nondegenerate]] [[skew-symmetric bilinear form]]. It is a basic result in [[linear algebra]] that any two such matrices differ from each other by a [[change of basis]].
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| The most common alternative to the standard Ω given above is the [[block diagonal]] form
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| :<math>\Omega = \begin{bmatrix}
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| \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
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| & \ddots & \\
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| 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}
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| \end{bmatrix}.</math>
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| This choice differs from the previous one by a [[permutation]] of basis vectors.
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| Sometimes the notation ''J'' is used instead of Ω for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a [[linear complex structure|complex structure]], which often has the same coordinate expression as Ω but represents a very different structure. A complex structure ''J'' is the coordinate representation of a linear transformation that squares to −1, whereas Ω is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which ''J'' is not skew-symmetric or Ω does not square to −1.
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| Given a [[hermitian structure]] on a vector space, ''J'' and Ω are related via
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| :<math>\Omega_{ab} = -g_{ac}{J^c}_b</math>
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| where <math>g_{ac}</math> is the [[metric tensor|metric]]. That ''J'' and Ω usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric ''g'' is usually the identity matrix.
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| ==Complex matrices==
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| If instead ''M'' is a ''2n''×''2n'' [[matrix (mathematics)|matrix]] with [[complex number|complex]] entries, the definition is not standard throughout the literature. Many authors <ref>{{cite journal|last = Xu|first= H. G.|title= An SVD-like matrix decomposition and its applications|journal= Linear Algebra and its Applications|date= July 15, 2003|volume= 368|pages=1–24|doi = 10.1016/S0024-3795(03)00370-7}}</ref> adjust the definition above to
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| {{NumBlk|:|<math>M^* \Omega M = \Omega\,.</math>|{{EquationRef|2}}}}
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| where ''M<sup>*</sup>'' denotes the [[conjugate transpose]] of ''M''. In this case, the determinant may not be 1, but will have [[absolute value]] 1. In the 2×2 case (''n''=1), ''M'' will be the product of a real symplectic matrix and a complex number of absolute value 1.
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| Other authors <ref>{{Cite journal|last1=Mackey |last2= Mackey|first1= D. S. |first2= N.|title= On the Determinant of Symplectic Matrices|year= 2003
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| |series=Numerical Analysis Report| volume= 422|publisher=Manchester Centre for Computational Mathematics|location=Manchester, England
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| }}</ref> retain the definition ({{EquationNote|1}}) for complex matrices and call matrices satisfying ({{EquationNote|2}}) ''conjugate symplectic''.
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| ==See also==
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| {{Portal|Mathematics}}
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| * [[symplectic vector space]]
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| * [[symplectic group]]
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| * [[symplectic representation]]
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| * [[orthogonal matrix]]
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| * [[unitary matrix]]
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| * [[Hamiltonian mechanics]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{planetmath reference|id=4140|title=Symplectic matrix}}
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| * {{planetmath reference|id=7455|title=The characteristic polynomial of a symplectic matrix is a reciprocal polynomial}}
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| [[Category:Matrices]]
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| [[Category:Symplectic geometry]]
| |
Msvcr71.dll is an important file that assists help Windows process different components of the program including important files. Specifically, the file is selected to help run corresponding files in the "Virtual C Runtime Library". These files are significant in accessing any settings which support the different applications plus programs in the system. The msvcr71.dll file fulfills numerous important functions; nevertheless it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a hard time processing plus reading components of the system. Nonetheless, consumers want not panic because this issue can be solved by following several procedures. And I can show you several tricks about Msvcr71.dll.
You may discover that there are registry products that are free and those that you'll have to pay a nominal sum for. Some registry products provide a bare bones program for free with the choice of upgrading to a more advanced, efficient adaptation of the same program.
So, this advanced double scan is not merely 1 of the greater, nevertheless it is also freeware. And as of all of this that many regard CCleaner one of the greater registry cleaners in the marketplace now. I would add that I personally like Regcure for the easy reason that it has a greater interface and I learn for a fact that it is ad-ware without charge.
Fixing tcpip.sys blue screen is easy to do with registry repair software.Trying to fix windows blue screen error on a own may be challenging considering should you remove or damage the registry it may cause serious damage to your computer. The registry should be cleaned and all erroneous and incomplete info removed to stop blue screen errors from occurring.The benefit of registry repair software is not limited to simply getting rid of the blue screen on business.We may be amazed at the greater and more improved speed and performance of the computer system following registry cleaning is done. Registry cleaning will really develop a computer's functioning abilities, especially when you choose a certain registry repair software that is especially efficient.
Another common cause of PC slow down is a corrupt registry. The registry is a pretty important component of computers running on Windows platform. When this gets corrupted your PC will slowdown, or worse, not commence at all. Fixing the registry is easy with all the employ of the program and registry mechanic.
Let's start with all the negative sides first. The initial price of the product is fairly inexpensive. However, it only comes with one year of updates. After which you need to subscribe to monthly updates. The advantage of that is the fact that ideal optimizer has enough funds plus resources to analysis mistakes. This way, we are ensured of safe fixes.
Reboot PC - Simply reboot a PC to see if the error is gone. Frequently, rebooting the PC readjusts the internal settings plus software plus therefore fixes the problem. If it doesn't then move on to follow the instructions under.
You are able to click here to locate out how to speed up Windows and strengthen PC perfomance. And you can click here to download a registry cleaner to aid you clean up registry.