# 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: $\det(O)=-1$ ...
1 vote
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 $\lambda$ ...
• 75
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 ...
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### 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 ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
• 1,433
1 vote