Talk:Symplectic matrix

J vs. Ω

On 19 april 2005, User:Fropuff changed the notation from J to Ω to "avoid confusion with complex structure". I rather want to change the notation back; we want to confuse the two, right? Unless we're trying to reserve J for the 2x2 case only, and Ω for the nxn case? linas 23:35, 18 March 2006 (UTC)[reply]

Never mind, I just hacked this article to say that the 2x2 matrix is called J. I wanted to link to this article from an article having J in it. linas 23:41, 18 March 2006 (UTC)[reply]

No, we don't! These are two very different things and should be distinguished. In particular, one could easily choose a basis for which Ω² ≠ −1, whereas this is an essential quality of a complex structure. Moreover, J should be understood as a linear transformation wheres Ω is a bilinear form. Given a hermitian structure on a vector space, J and Ω are related via

\Omega _{ab}=-g_{ac}{J^{c}}_{b}

That J and Ω have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that g is usually to be the identity matrix. -- Fropuff 23:57, 18 March 2006 (UTC)[reply]

Excellent point. Since the distinction is all too easily missed, I copied this into the article. FYI, the article on Hermitian manifold defines this, but with a lower-case &oemga; instead. (and no minus sign). 67.100.217.178 04:33, 30 March 2006 (UTC)[reply]

Opening lines with definition are ambiguous. It is left unclear what is the actual requirement of matrix Ω in the rest of the article: does the sentence "Note that Ω has determinant +1 and has an inverse giv...." refer to the general non-singular, skew-symmetric Ω, or just to the cases of Ω following the "Typically..."? Can this be clarified? (All 2n×2n, nonsingular, skew-symmetric matrices have determinant >0, but only those of these with orthonormal columns possess the other parts of the property. Are we to assume that all cases under discussion, not only the "Typically..." ones, also use an Ω matrix that has orthonormal columns?) —Preceding unsigned comment added by 83.217.170.175 (talk) 21:21, 29 March 2010 (UTC)[reply]

definition

In the case of complex matrix M, is $M^{T}$ replaced by $M^{*}$ (conjugate transpose) as is usually the case with complex analogues for real matrices? Just a small clarification that the article needs. 71.208.184.209 (talk) 06:12, 8 November 2010 (UTC)[reply]

Overall quality and merger

In its present form this article is not helpful and mathematically not sound. The dimension of a group is not well defined and the statement "An example of a group of symplectic matrices is the group of three symplectic 2x2-matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1." is not correct. In view of the existence of an article on the symplectic group, I'd say one should merge the two. — Preceding unsigned comment added by 62.198.81.12 (talk) 20:20, 24 October 2016 (UTC)[reply]

Ferraro et. al. 2005 Section 1.3.

Is it book, or article? Do this article have title? Jumpow (talk) 21:17, 9 May 2017 (UTC)[reply]

We probably should ask UtherSB (talk · contribs), I think he added the text that uses that reference (see diff).

For what it's worth, the article Symplectic group references

Ferraro, Alessandro; Olivares, Stefano; Paris, Matteo G. A. (March 2005), Gaussian states in continuous variable quantum information, arXiv.org.

Probably the same source. – Tea2min (talk) 05:38, 10 May 2017 (UTC)[reply]

Essential question needs to be addressed

The article defines a symplectic matrix in terms of a fixed choice of skew-symmetric matrix 𝝮. It also mentions a common choice for 𝝮, and later in the article mentions a common alternative choice for 𝝮.

But the article never addresses the essential question of whether the concept of a symplectic matrix depends on the choice of 𝝮, or whether the same set of matrices is defined to be symplectic regardless of which skew-symmetric matrix 𝝮 is used in the definition.

I hope someone knowledgeable on this subject will address this essential question in the article.71.37.182.254 (talk) 19:28, 9 November 2020 (UTC)[reply]

The answer is "yes, it depends"; this is addressed obliquely in the (completely unsourced) section Symplectic_matrix#The_matrix_Ω. It would be good to find sources for that section, which might incidentally guide making the answer to your question more explicit in the text. --JBL (talk) 19:44, 9 November 2020 (UTC)[reply]

Simpler proof of determinant of Symplectic matrix equals 1

I was wondering if the editors would consider adding a "simpler" proof of the fact that symplectic matrices have determinant equal to 1. Currently, the argument relies on invoking the Pfaffian. Here is an argument that although longer requires only basic notions of linear algebra.

1. Note that the Polar decomposition of symplectic matrix gives a positive definite symplectic matrix and an orthogonal symplectic matrix. For a proof see e.g. Eqns. 23 and 24 of https://cdnsciencepub.com/doi/pdf/10.1139/cjp-2024-0070 [Although this is also stated in the section Diagonalization and Decomposition]. We can then write S= U R with U symplectic and orthogonal and R symplectic and positive-definite. From this it follows that det(S) = det(U) det (R) = det(U) since we know det(R)=1 as it is Symplectic and positive-definite.

2. It is also easy to show (cf. Appendix B1 of Serafini https://www.taylorfrancis.com/books/mono/10.1201/9781315118727/quantum-continuous-variables-alessio-serafini) that every symplectic orthogonal matrix has the form $U={\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&I_{n}\\-iI_{n}&iI_{n}\end{pmatrix}}{\begin{pmatrix}V&0_{n}\\0_{n}&V^{*}\end{pmatrix}}\left[{\frac {1}{\sqrt {2}}}{\begin{pmatrix}I_{n}&I_{n}\\-iI_{n}&iI_{n}\end{pmatrix}}\right]^{\dagger }$ where V is a unitary matrix of size n. Taking determinants on both sides gives det(U) = det(V) det(V^*). From this it follows that the det(U)=1 and thus det(S) = 1.

I should also mention that I believe the Proof of Proposition 34 of Gosson's "Symplectic Methods in Harmonic Analysis and in Mathematical Physics" has a minor typo. They write, for the positive part of the Polar decomposition of a Symplectic S to be $R=SS^{T}$ where it should be $R=[SS^{T}]^{1/2}$ . Moreover, the current reference 5 by Ferraro, Olivares and Paris only states the 'Euler' or 'Bloch-Messiah' decomposition of a Symplectic matrix. A simple proof can be found in https://cdnsciencepub.com/doi/pdf/10.1139/cjp-2024-0070 Nicolas.quesada (talk) 02:07, 16 December 2024 (UTC)[reply]