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Questions tagged [symplectic-linear-algebra]

Questions about vector spaces equipped with a symplectic form, a non-degenerate, skew-symmetric bilinear form.

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$O$ orthogonal with $\det(O)=-1$ implies $||\Omega - O \Omega O^{T}|| = 2 $?

Let $O\in \mathrm{O}(2n)$ be an orthogonal matrix. Let $\Omega$ be the matrix $\Omega:= \bigoplus^n_{i=1} \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}.$ Is it true that: $\det(O)=-1$ ...
Dante Perès 's user avatar
1 vote
1 answer
70 views

Prove that if $\lambda$ is an eigenvalue of a symplectic matrix, then $\frac{1}{\lambda}$ is also an eigenvalue of such matrix

I´m trying to solve the following problem: A symplectic $n\times n$ matrix $A$ follows this conditions: $J$ is a $n\times n$ matrix $J^2=-I$ $A^TJA=J$ $n$ is an even number Prove that if $\lambda$ ...
gnzlama's user avatar
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0 answers
24 views

Find a Lagrangian subspace complementary to two subspaces

Let $(V,\omega)$ be a symplectic vector space and $A,B$ be two subspaces of $V$ with $2\dim A=2\dim B=\dim V$. I need to prove that there always exists a Lagrangian subspace $L$ of $(V,\omega)$ that ...
FUUNK1000's user avatar
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Symplectic transformation for which componentwise area is not conserved in phase space

Consider two elements the phase space $x, y\in \mathbb{R}^{2n}$. Denote the coordinates $x=(x_q,x_p)=(x_q^1,\ldots, x_q^n,x_p^1,\ldots, x_p^n)$ and the same for $y$. If we consider the bilinear form $$...
edamondo's user avatar
  • 1,327
1 vote
0 answers
61 views

Symplectic approximation to a given matrix

I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
Dante Perès 's user avatar
0 votes
1 answer
27 views

$\mathfrak{sp}_{2n}$-stable spaces

Let $\mathfrak{sp}_{2n}(\mathbb{F})$ be the symplectic Lie algebra over an algebraic closed field $\mathbb{F}$ of characteristic $0$, that is, the space $\{X\in M_{2n}(\mathbb{F}) | X^T\Omega+\Omega X ...
Fernando Nazario's user avatar
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70 views

Lagrangian invariant subspace of symplectic matrix

Suppose $S$ is symplectic matrix with only real eigenvalues. I need to prove that $S$ has Lagrangian invariant subspace, i.e. there is $L$ - Lagrangian, such that $S(L) \subset L$. I know that ...
User2035's user avatar
0 votes
0 answers
36 views

Symplectic Area vs. the area of a Riemannian parallelogram.

Given two vectors $u,v\in\mathbb{R}^2$ and a Symplectic form $\omega$ with compatible inner product $g$, define the Symplectic area of $u,v$ as $\omega(u,v)$ and the Riemannian area to be the area of ...
abc2003's user avatar
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1 answer
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linear complex structure as submanifold of general linear group

Given a finite dimensional real vector space, we can view all the complex structures on it as a subspace of ${\rm GL}(V)$. I wonder if it is a submanifold of ${\rm GL}(V)$. And moreover, given a ...
Qhejaz's user avatar
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Sesquilinear generalization of symplectic form

I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form. Fix $\mathbb K$ to be a field, and $...
Cole Comfort's user avatar
1 vote
0 answers
86 views

Decompositions of symplectic matrices over the integers

Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with $$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$ what known decompositions exist such ...
Cameron's user avatar
  • 429
1 vote
1 answer
82 views

degeneracy of even-dimensional "pullback" of symplectic form

Suppose $\omega(u,v) = \left\langle Ju, v\right\rangle$ is the canonical nondegenerate symplectic form on $\mathbb{R}^{2N}$. Given a random left-orthogonal matrix $P\in\mathbb{R}^{2N\times 2n}$ (...
MathIsArt's user avatar
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1 answer
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Alternating Frobenius form in Sage

Given an alternating, non-degenerate matrix $A$ over the integers, I need to compute the matrix "closest" to the standard symplectic form that can be obtained from $A$ by an integer change ...
Oliver Miller's user avatar
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0 answers
33 views

Construction of the Adelic Metaplectic Group

I'm essentially looking for a reference for the following statement on the wikipedia: It can be proved that if F is any local field other than $\mathbb{C}$, then the symplectic group $Sp_{2n}(F)$ ...
Riley Moriss's user avatar
1 vote
0 answers
60 views

Modified version of Darboux's theorem

Say we have a 4 dimensional real manifold and two 2-forms, called $k$ and $\omega$. Assume $$ d k = 0, \hspace{1 cm} d \omega = 0. $$ Furthermore, assume $\omega$ is non-degenderate, but $k$ is not. ...
user1379857's user avatar
3 votes
0 answers
41 views

Linear Legendrian relations

There is a "partial category" of Lagrangian relations, otherwise known as canonical relations, whose objects are symplectic vector spaces and whose morphisms are Lagrangian submanifolds. By ...
Cole Comfort's user avatar
1 vote
0 answers
29 views

Reference for Cardinality of Parabolic Subgroup of Symplectic Group over Finite Fields

I am looking for a source to reference that gives the cardinality of parabolic subgroups of the Symplectic group $Sp$ over a finite field $\mathbb F_q$. What I want is essentially exactly what is in ...
Ryan L's user avatar
  • 21
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1 answer
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Difference between relative Moser and Weinstein's Lagrangian neighbourhood theorem

The statement of relative Moser: Let M be a manifold, X a compact submanifold, i : X → M the inclusion map, and $ω_0$ and $ω_1$ symplectic forms on M such that $i^\ast ω_0 = i^\ast ω_1 $. Then there ...
Abdullah Ahmed's user avatar
2 votes
0 answers
87 views

Existence of Orthogonal Symplectic matrix

I am investigating the existence of an orthogonal matrix satisfying specific conditions. Let $\mathbf{v}_1, \dots, \mathbf{v}_M \in \mathbb{R}^{2n}$ be real vectors with unit norm, where $M \leq 2n$. ...
Dante Perès 's user avatar
1 vote
1 answer
127 views

Non-degenerate $2$-form on a manifold gives tangent-cotangent bundle isomorphism

I am trying to solve the following problem: Let $M$ be a smooth $n$-manifold, and let $\omega\in \Omega^2(M)$ be such that $\omega_p\colon T_pM\times T_pM\to \Bbb R$ is non-degenerate for each $p\in ...
Random's user avatar
  • 629
1 vote
0 answers
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A step in the Gromov's proof of contractability of $\omega$-tame complex structures on a finite dimensional vector space

Given a symplectic form $\omega$ on a symplectic vector space $V$, a complex structure $J$ on $V$ is said to be tamed by $\omega$ or $\omega$- tamed if $$\omega(v,Jv)>0$$ for all non zero $v\in V$. ...
Uncool's user avatar
  • 940
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The topology of symplectic group Sp(2n)

I am reading V. I. Arnol’d, A. B. Givental’ , Symplectic Geometry $\S$4.4 about the topology of the symplectic group Sp(2n). It is stated that The manifold $Sp(2n, \mathbb{R})$ is diffeomorphic to ...
Mr. Egg's user avatar
  • 638
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Subgroup of $Sp(2n,\mathbb{R})$.

I am stuck on this problem and cannot seem to find a good reason for drawing the required conclusion. The problem is as follows: I know that the maximal dimension of abelian Lie subalgebra of $Lie(Sp(...
Yushi MuGiwara's user avatar
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0 answers
47 views

Show that for any $L_1, L_2\in Lag(E,\omega)$ there exists a symplectic basis in which $L_1=<e_1,..,e_n>$ and $L_2 =<e_1,...,e_k,f_{k+1},..,f_n>$

I'm a beginner on sympletic geometry, studiyng by the Meinrenken notes, need a help in a question. Let $(E,\omega)$ sympletic space and $L_1, L_2\in Lag(E,\omega)$ lagrangians such that $dim(L_1\cap ...
Gabriel Costa's user avatar
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0 answers
37 views

About n-dimensional subspace of a 2n-dimensional symplectic vector space

I was working on a question that asks me to determine whether it is possible for a 2-dimensional subspace in a 4-dimensional symplectic vector space is neither symplectic nor Lagrangian, as well as ...
Runyang Wang's user avatar
3 votes
0 answers
240 views

Preserving the symplectic 2-form vs phase space volume

Say I have a Hamiltonian system of $N$ particles in 3D-3V phase space. I'm using some sort of update scheme taking the system from $t^{n-1}$ to $t^{n}$ to $t^{n+1}$. I want to know if the update ...
Grey Haven's user avatar
2 votes
0 answers
60 views

On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.2-3. (symplectic map is immersion)

I'm either confused with the definition of symplectic forms / immersions or the way exercise 5.2-3 in Marsden's 'Introduction to Mechanics and Symmetry' was stated. It reads as follows Exercise 5.2-3....
Alfons Winkel's user avatar
0 votes
0 answers
91 views

Decomposing binary matrices

Given a $2n \times 2n$ binary symplectic matrix$^1$ $M$, I need to decompose that into a product of matrices from the following set S. It is guaranteed that multiple such decompositions exist. $S = \{ ...
FDGod's user avatar
  • 101
3 votes
1 answer
50 views

Sum of symplectic complement subspaces closed?

I have the following problem (p.73 in Marsden's 'Introduction to Mechanics and Symmetry' Book): Given is an infinite-dimensional Banach space $Z$ and on it a weakly non-degenerate, symplectic form $\...
Alfons Winkel's user avatar
1 vote
1 answer
73 views

general form of group elements of the symplectic group

For the unitary group $U(n)$, its group elements are of the form $e^{iH}$, with $H$ being a $n\times n $ hermitian matrix. Do we have a similar expression for the group element of the symplectic group ...
poisson's user avatar
  • 1,015
2 votes
0 answers
52 views

How does a totally complex polarization induce a Kähler structure of the symplectic manifold

A totally complex polarization of a symplectic manifold $(M,\omega)$ is a subbundle $F$ of the complexified tangent bundle $TM_{\mathbb C}:=TM\oplus iTM$ such that $F$ is integrable in the sense that ...
Kanae Shinjo's user avatar
1 vote
1 answer
72 views

What is the geometry of $L=\lbrace W \subset V:\text{$W$ is Lagrange subspace of $V$} \rbrace$?

I know the following statement: Theorem. Let $(V,\omega)$ be a finite dimensional symplectic linear space. Then the symplectic group $Sp(V)$ of $V$ transitively acts on the set $L=\lbrace W \subset V:...
s.h's user avatar
  • 476
3 votes
0 answers
62 views

Help finding specific isomorphism between $\mathbb{Q}^{2g}$ and $K^2$.

I'm reading McMullen's paper "Billiards and Teichmüller Curves on Hilbert Modular Surfaces." I am stuck on understanding isomorphism (6.2) in the "Hilbert modular varieties" ...
Sam Freedman's user avatar
  • 3,989
0 votes
1 answer
62 views

How do I complete this prove of $\Phi$ is linear and symplectic iff its graph is a Lagrangian subspace of $(V \times V,(−\omega) \oplus \omega)$?

Let $(V, \omega)$ be a symplectic vector space. Prove that a map $\Phi : V \mapsto V $is linear and symplectic if and only if its graph $\{(v, \Phi v)| v \in V\}$ is a Lagrangian subspace of $V \times ...
some_math_guy's user avatar
1 vote
1 answer
62 views

Proving non degeneracy of symplectic form over the linear symplectic quotient

$(V, \omega)$ symplectic vector space a$W \subseteq V$ linear subspace. We define the reduced space $(\bar W , \bar \omega)$ as $\bar W = W/(W \cap W^\omega)$, equipped with the form $\bar\omega ([...
some_math_guy's user avatar
0 votes
1 answer
89 views

Prove $\dim {\rm Im}\ T = \dim W$ in the proof of the dimension theorem for the symplectic complement

I am trying to prove the dimension theorem for the symplectic complement and I am missing a small step right at the end: Let $ (V,\omega) $ be a symplectic vector space and $W$ a linear subspace of $V$...
some_math_guy's user avatar
1 vote
0 answers
34 views

Lagrangian Grassmanian as Homogeneous Space

This is part of Exercise 2.14 in Kirillov's "An Introduction to Lie Groups and Lie algebras". The reader is supposed to prove that $Sp(2n,\mathbb{R})$ acts transitively on the space $L_n$ of ...
topolosaurus's user avatar
  • 1,943
1 vote
1 answer
54 views

Bound quotient of maximal and minimal singular values

I am working on a problem where the following quantity emerges: $$ \frac{\sigma_{\text{max}}(J-M)}{\sigma_{\text{min}}(J+M)} $$ where $J$ is the canonical symplectic matrix, $M^T=M$ and $M$ is ...
Dadeslam's user avatar
  • 826
0 votes
1 answer
160 views

Proving that $\operatorname{Sp} (n)$ is a group

I'm trying to prove that the symplectic group is, indeed, a group. It was easy to show that the operation is closed and the neutral element is inside the group itself, but I'm stuck trying to show ...
ErikLAndre's user avatar
6 votes
0 answers
141 views

If $A\geq S^TS$ and $A\geq \lambda$ for symplectic $S$, is $A\geq R^TR\geq \lambda$ for some symplectic $R$?

Consider a $2n$-by-$2n$ real matrix $A$ such that there exists symplectic matrix $S \in Sp(2n,\mathbb{R})$ with $A \geq S^TS$ and $A \geq \lambda \mathbb{1}$ for some $\lambda \in [0,1]$. Does there ...
Daniel Ranard's user avatar
0 votes
1 answer
64 views

Order of Symplectic Matrices

I'm going through Appendix 3 of Lax's Linear Algebra, and I'm not entirely sure why symplectic matrices must be of order $2n$. He defines symplectic matrices as "linear maps that preserve a ...
Redcrazyguy's user avatar
0 votes
2 answers
135 views

Stabiliser group [closed]

Let $G=\operatorname{Sp}_{2r}(2)$. There are two orbits of $G$ on the natural $G$-set, one having the identity, the other having all the remaining elements. What is the subgroup of $G$ that stabilizes ...
scsnm's user avatar
  • 1,283
3 votes
1 answer
58 views

Characteristic foliation is a rank 1 subbundle

I don't see why the following fact is true, although I thought it is purely a linear algebra problem. Let $(M^{2n},\omega)$ be a symplectic manifold and $Y \subset M$ be a $(2n-1)$ hypersurface. We ...
Yoona's user avatar
  • 421
0 votes
0 answers
78 views

Inner Product of a Symplectic Matrix

I'm going through appendix 3 of Lax's Linear Algebra book, and I'm a bit confused about the concept of the symplectic matrix. From what I understand, we are looking for a matrix $A$ which fulfills $(...
Redcrazyguy's user avatar
3 votes
1 answer
406 views

Why is an area preserving diffeomorphism a symplectomorphism (in $R^2$)

Given this very simple sympletic vector space: $\left(\mathbb{R}^2, \mathrm{~d} x \wedge \mathrm{d} y\right)$, how can we show that an area preserving diffeomorphism $f: \mathbb{R}^2 \rightarrow \...
Bill's user avatar
  • 4,441
0 votes
1 answer
58 views

What is the point of the sympletic lie algebra [closed]

I have taken a lie algebra class this semester, and have stumbled upon a particularly unmotivated lie algebra, the sympleitc lie algebra. So my question is quite simple, why is this an interesting ...
DevVorb's user avatar
  • 1,433
1 vote
1 answer
30 views

Given $L$ Lagrangian and $M$ isotropic with $ M\cap L=\{0\}$, show there exists $e\in M^\sigma\smallsetminus(L+M)$

In this post: Passage in proving lagrangian subspaces have lagrangian complements where the poster wants to prove that Lagrangian subspace has Lagrangian complement, one of the intermediate steps is ...
Bill's user avatar
  • 4,441
2 votes
2 answers
386 views

How to define the Symplectic form on the cotangent bundle of a Lie group?

Let $G$ be a Lie group, for the convenience of computation, you can assume $G=\mathrm{GL}_n(\mathbb{R})$, its cotangent bundle $T^*G$, by the left trivialisation, is actually a trivial bundle $T^*G\...
Z. Liu's user avatar
  • 458
1 vote
2 answers
85 views

A property of the wedge of the restriction of a $2$-form and a $1$-form to a codimension $1$ vector subspace

Let $\omega \in \Lambda^2(V^*)$ be a $2$-form on a vector space $V$ of dimension $2k+2$. Suppose further we have a nonzero $1$-form $\eta$ on $V$ and a subspace $W$ of codimension $1$ with the ...
rosecabbage's user avatar
  • 1,665
1 vote
1 answer
45 views

Decomposition of a finite symplectic abelian group into hyperbolic subgroups

If $G$ is a non degenerate symplectic group, we can decompose it as an orthogonal direct sum of hyperbolic subgroups $G_i$ (meaning non degenerated symplectic of rank 2). See for instance "...
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