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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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Unitary group acts transitively on the Lagrangian Grassmannian

I am trying to prove that the unitary group associated to an $\omega$-compatible complex structure $J$ acts transitively on the Lagrangian Grassmannian $\mathcal{L}(V)$. I know that for a symplectic ...
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15 views

Williamson's theorem for positive semi-definite matrices of even size

I know the Williamson's theorem for positive definite matrices of even size. I was wondering if the theorem also holds for positive semi-definite matrices with even rank. More precisely, if $A$ is a $...
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90 views

A description of the compact symplectic group

Let $\mathrm{Sp}(2m;\mathbb{C})=\{X\in\mathrm{GL}(2m;\mathbb{C});X^t\Omega X=\Omega\}$, where $\Omega=\begin{bmatrix}0& I_m\\ -I_m& 0\end{bmatrix}$, $I_m$ is $m\times m$ identity matrix. The ...
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1answer
158 views

Quadratic first integrals

I started reading chapter II.16 of Solving Ordinary Differential Equations I in order to understand nummerical methods more. There, they state that symplectic methods don't always conserve the first ...
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1answer
65 views

$SP_{2n}(\mathbb {R})$ acts transitively on $\mathbb {R}^{2n}$

I am trying to prove that $SP_{2n}(\mathbb {R})$ acts transitively on $\mathbb{R}^{2n}$ by $(A,x)=Ax$, where $SP_{2n}(\mathbb {R})=\{A\in GL_{2n} (\mathbb{R})| A^{T}J_{2n}A=J_{2n}\}$ is the real ...
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1answer
33 views

Are all symplectic $(0,1)$-matrices lower/upper block-triangular?

Context. I don't expect this question is actually interesting, it just seems like a nice/fun exercise to get better acquainted with $Sp(2n)$ (and to celebrate the new year!). In this question ...
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1answer
36 views

subspaces of a symplectic vector spaces are of special forms.

Let $(V,\omega)$ be a symplectic vector space. Let $F \subseteq V$ be a subspace. Show that $V$ admits a symplectic basis $\{e_1,\ldots,e_n,f_1,\ldots,f_n\}$ with the following properties: (1) If $...
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29 views

Topological vector space completion with respect to a symplectic form?

Suppose we have an infinite-dimensional vector space $V$ with a symplectic form $\omega:V\times V\to\mathbb R$. It can be given a weak topology that makes $\omega$ continuous. Does it make sense to ...
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43 views

The defining matrix of a symplectic matrix

Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me. A $2n\times 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition: $$M^T\...
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175 views

Determinant of matrix formed from blocks of a $2 \times 2$ block partitioned symplectic matrix.

While working on a problem in quantum optics, I came across the following determinant of a complex matrix of size $n \times n$ : $$\mathbb{G}=\det\left[\mathcal{U}_{11}^{}+\mathcal{U}_{12}^{}\mathcal{...
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Invariant homogeneous polynomial on quaternions

Let $\mathbb{H}$ denote the quaternions. If $(w_1,\ldots,w_n)\in \mathbb{H}^n$ we can write $w_i=z_i+jz_{n+i}$ with complex numbers $z_1,\ldots,z_{2n}$. Now let $M$ be the group of all matrices of the ...
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1answer
50 views

Symplectic forms are isomorphic

Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem: All symplectic forms on $V$ are ...
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Symplectic reduction of a linear symplectic space.

I want to know what is the symplectic reduction of a symplectic linear space. Suppose $(V,\omega)$ is vector space with a nondegenerate bilinear form $\omega$. We can assume it is $(\mathbb R^{2n},\...
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1answer
366 views

Diagonalization of symplectic matrix

I tripped over the following assertion. Let $M \in \mathrm{Sp}(2n)$ be a symplectic real matrix which is diagonalizable. Then we can write it down as $M = S^{-1}D\ S$ where $S \in \mathrm{Sp}(2n)$ ...
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1answer
137 views

Sp(2n, R) = SL(2n, R)

I have proved that Sp(2n,R) is a subgroup of SL(2n,R). But is there an equality? If no, what counter example can do we have? Thanks
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50 views

How can I verify that this standard map is symplectic?

Show that the following standard map is symplectic $$ I(t+1) = I(t) + K\sin\theta(t) $$ $$ \theta(t+1) = \theta(t) + I(t+1) \bmod 2 \pi$$
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Linear symplectic reduction

I am studying this article: BC: Oliver Baues, Vicente Cortès - Symplectic Lie Groups I-III. It is my first step in studying simplectic geometry. I have great difficulty to prove [Lemma 1.1.][BC:] ...
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“Cauchy-Schwartz” for symplectic forms

I would like to show some sort of "Cauchy-Schwartz" inequality for symplectic maps. i.e. given a symplectic map $\phi:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ and $u := \phi(e_1),v:=\phi(f_1)$, ...
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1answer
30 views

Lagrangian subspace is preserved by almost complex structure

I have a Lagrangian subspace $L$ of $\mathbb{R}^{2n}$ (i.e. $L^{\perp_{\omega_0}} = L$ where $\omega_0$ is the natural symplectic form. I want to show that, given the natural almost complex structure ...
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1answer
43 views

Antisymmetric Bilinear Forms and Wedge Products

I want to show that every antisymmetric bilinear form on $\mathbb{R^3}$ is a wedge product of two vectors. In other words, suppose we have a basis $\vec{u}$, $\vec{v}$, and $\vec{w}$ for $\mathbb{R}^{...
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Coordinate-free proof of non-degeneracy of symplectic form on cotangent bundle

It's relatively straightforward to provide a coordinate-free definition of the symplectic form on a cotangent bundle; the usual way to do this is to construct the tautological 1-form $$\lambda(\xi) = \...
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1answer
61 views

Direct product of group with itself mod diagonal subgroup

Let $G$ be any abelian group, and let $\triangle_G = \{(g,g)\mid g\in G\}\subset G\times G.$ Is there any significance in studying the quotient group ${(G\times G)}\left/{\triangle_G}\right.?$ If ...
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1answer
153 views

Finite dimensional symplectic vector space over field of characteristic 2 has even dimension

Let $V$ be a finite dimensional over an arbitrary field $F$ and $f: V \times V \to F$ be a symplectic form, meaning f is bilinear, f is alternating, i.e. $f(v,v)=0$ for all $v \in V$, and f is non-...
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1answer
31 views

Antisymmetric Bilinear Forms

What would be an antisymmetric bilinear form on $\mathbb{R^{4}}$ that cannot be written as a wedge product of two covectors? Also, what would be a $4\times4$ skew-symmetric matrix of rank $4$? Finally,...
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hyperplane in symplectic vector space

I found the following reasoning that, it seems immediate in the document, but for me it is not. It comes from tangent spaces of codimension 1 submanifolds on a symplectic manifold, but it can be ...
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2answers
27 views

Action of $\operatorname{GL}(V)$ on $\mathcal J(V)$ is transitive

Let $V$ be vector space with a complex structure $J$. Define a map $\Psi\colon\operatorname{GL}(V)\longrightarrow\mathcal J(V)$ where $\Psi(\Phi)=\Phi^{-1}J\Phi$ where $\mathcal J(V)$ is the group of ...
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1answer
58 views

Is $\mathbb{T}^3\times \mathbb{S^3}$ symplectic?

I'm trying to verify whether or not $M:=\mathbb{T}^3\times \mathbb{S}^3$ is a symplectic manifold. Here is my attempt: first, I've noticed that $\mathbb{S}^1\times \mathbb{S}^3$ is not symplectic, ...
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69 views

How do I find canonical coordinates for the Lorentz group generators?

Consider the Poisson brackets (symplectic structure) given by the Lorentz algebra (Lie algebra of $SO(1,3)$) $$\{M^{AB},M^{CD}\} \equiv \omega_{AB,CD}\mathrm{d}M^{AB} \mathrm{d} M^{CD} = \eta^{AC} M^{...
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1answer
72 views

Kernel of a symplectic form is one dimensional

I was reading through "Lectures on Symplectic Geometry" by Ana Cannas da Silva. In Chapter 10 where the book introduces contact structures, it says $ker \ (d\alpha_p)$ is one dimensional where $\...
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1answer
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link between symplectic characterisation and isotropic or symplectic subspaces

I've stumbled on this exercise I have a symplectic capacity $c$ and $O$ an open set of $\mathbb{R}^{2n}$ and $W$ a $2n-2$ dimensional subspace $W$ of $\mathbb{R}^{2n}$. I have to show that $c(O+W) = \...
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1answer
447 views

Non-degenerate differential forms

I'm looking into symplectic forms which requires the differential 2-form to be closed and non-degenerate. I know that in Euclidean space, a differential 2-form $\omega:\mathbb{R}^n\times \mathbb{R}^n\...
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1answer
113 views

Every $E\subset V$ with $\dim E=\frac{1}{2}\dim V$ has a Lagrangian complement

Let $(V, \omega)$ be a symplectic $2n$-dimensional vector space and $E\subset V$ any $n$-dimensional subspace. Prove there is a Lagrangian subspace $L\subset V$ with $V=E\oplus L$. [here, Lagrangian ...
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Bilinear forms by P B Battacharya, Linear Algebra, Chap 7 Example 7.1.10.(4)

Let B be a bilinear form on a vector space V over R whose matrix wrt basis (e1,e2) of V is ((1,2),(2,-1)) row-wise. Find a new basis of V wrt which the matrix of B is identity matrix. How to proceed ?...
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104 views

$W_1,…,W_k\subset V$ lagrangian subspaces $\Rightarrow \exists\, L\subset V$ lagrangian with $L\cap W_i=\{0\}$ for all $i$

Let $(V,\omega)$ be a symplectic vector space and $W_1,...,W_k\subset V$ Lagrangian subspaces. Prove there is a Lagrangian subspace $L\subset V$ such that $L\cap W_i=\{0\}$ for all $i$. I've noticed ...
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178 views

Geometric meaning of Liouville vector field.

A vector field $v$ on a symplectic manifold $(X,\omega)$ is a Liouville vector field if $\mathcal{L}_v\omega=\omega.$ Can anyone explain a geometric meaning of it?
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103 views

Existence of natural symplectomorphism for two structures in $V \times V$.

Let $(V,\Omega)$ be a symplectic vector space. In the product $V \times V$, we have the following two symplectic structures: $$(\Omega \ominus \Omega)((x,u),(y,v)) = \Omega(x,y) - \Omega(u,v)$$and $$\...
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Gompf's symplectic sum construction and symplectic involution of annulus

In Gompf's 1994 article, "Some new symplectic 4-manifolds," the author defines a connect sum operation for two symplectic manifolds of the same dimension along a common codimension-2 symplectic ...
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Identifying a specific $\operatorname{Sp}(4,\mathbb{C})$-representation

Let $V$ be some given vector space and $\operatorname{Sp}(4,\mathbb{C})\rightarrow \operatorname{Gl}(V)$ be a finite dimensional representation. Let $H\subset \operatorname{Sp}(4,\mathbb{C})$ be the ...
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Adjoint orbit of an regular and elliptic element of $\mathrm{SP}(2n,\mathbb{R})$

Let $G = \mathrm{SP}(2n,\mathbb{R})$, $\mathfrak{g} = \mathfrak{sp}(2n,\mathbb{R})$, $\mathfrak{h} = \mathfrak{sp}(2m,\mathbb R) \leq \mathfrak{g}$ and $\mathfrak q= \mathfrak h^{\perp}$. We know ...
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83 views

Good free calculator for manipulating symbolic matrices of 6x6 and larger?

I have a symbolic 6-by-6 matrix containing the (pseudo-Riemannian) metric coefficients for a 6-dimensional manifold. I am trying to calculate the inverse matrix in service of working out the ...
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1answer
97 views

Symplectic form and quadratic form

Let $W = \mathbb{F}_{2^{m + 1}} \oplus \mathbb{F}_{2^{m + 1}}$ be a $2(m+1)$-dimensional vector space over $\mathbb{Z}_2$ equipped with a symplectic form $ \langle \cdot , \cdot \rangle : W \times W \...
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61 views

Question on constructing a smplectomorphism between $(V,\omega)$ and $(Y \oplus Y^*,\omega_0)$ where $Y$ is lagrangian

This is exercise 9, p.8 in Lectures on Symplectic Geometry by Ana Cannas da Silva. Let $(V,\omega)$ be a symplectic vector space and $Y$ a lagrangian subspace of $V$, i.e. $Y = Y^\omega$. On $Y \...
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vanishing of a certain cyclic permutation sum of tensors

I am stuck while reading what appears to be a very basic differential geometric argument about existence of torsion-free symplictic connections from here: https://arxiv.org/pdf/math/0511194.pdf It is ...
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1answer
42 views

$\mbox{PSL}(2,\mathbb{R})$ and $\mbox{Sp}(2,\mathbb{R})$

I am wondering if somebody could point me to some kind relationship between groups $\mbox{PSL}(2,\mathbb{R})$ and $\mbox{Sp}(2,\mathbb{R})$. I know very little about Lie Algebras and group ...
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1answer
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Determining two binary quadratic forms induced by binary nondegenerate symplectic form.

$V$ is a vector space over $\mathbb F_2$ of dimension $2n$. $B : V\times V \to \mathbb F_2$ is a nondegenerate symplectic form. I'm reading a paper that states that there are essentially two ...
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148 views

Quick question on symplectic vector spaces

Given a symplectic vector space $V$, and a subspace $ \ U \subset V$, if $ \ U^\bot \not\subset \ U$, ($U \ $ is not co-isotropic), is it necessarily the case that $U \subseteq U^\bot \ $ or $ \ U \...
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106 views

Lines in Grassmannian

Let $\{x_1,x_2,x_3,x_4,x_5,x_6\}$ be a basis of $\mathbb C^6$. Let $V \in \mathrm{Gr}(3,\mathbb C^6)$ be such that $$V=\left<\sum_{i=1}^6a_ix_i, \sum_{i=1}^6b_ix_i, \sum_{i=1}^6c_ix_i\right>$$ ...
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214 views

How do I derive the center of the symplectic group?

I define the real symplectic group to be the set of $2n \times 2n$ matrices satisfying \begin{equation} S\Omega S^\intercal = \Omega, \end{equation} where \begin{equation} \Omega = \bigoplus_{i=1}^n \...
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54 views

Show that $\mathfrak{sp}_{2n}(\mathbb{C})$ is a sub lie-algebra of $\mathfrak{gl}(\mathbb{C}^{2n})$

I want to show that $\mathfrak{sp}_{2n}(\mathbb{C})$ is a sub lie-algebra of $\mathfrak{gl}(\mathbb{C}^{2n})$. Where $$\mathfrak{sp}_{2n}(\mathbb{C})=\{X \in End(\mathbb{C}^{2n})| \Omega(Xv,w)+\Omega(...
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55 views

How does one define a non-degenerate form on a non-self-dual topological vector space?

Duality in this question means topological duality. One can define a non-degenerate form on a Hilbert space, since $H\cong H^*$. But this is not true for, say, a Schwartz space $S(\mathbb{R})$, since $...