Matrice hamiltonienne

En mathématiques, une matrice hamiltonienne (ou de Hamilton) A est une matrice réelle 2n×2n satisfaisant la condition que le produit KA soit symétrique, K étant la matrice antisymétrique :

K={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}

et I_n étant la matrice identité n×n. En d'autres termes, $A$ est hamiltonienne si et seulement si :

KA-A^{T}K^{T}=KA+A^{T}K=0.\,

Dans l'espace vectoriel des matrices 2n×2n, les matrices hamiltoniennes forment un sous-espace vectoriel de dimension 2n² + n.

Propriétés

Soit $M$ une matrice par bloc 2n×2n donnée par :
$M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}$

où $A,B,C,D$ sont des matrices n×n. Alors $M$ est une matrice hamiltonienne à condition que $B,C$ soient symétriques et que $A+D^{T}=0$ .
La transposée d'une matrice hamiltonienne est hamiltonienne.
La trace d'une matrice hamiltonienne est nulle.
Le commutateur de deux matrices hamiltoniennes est hamiltonien.
Les valeurs propres de $M$ sont symétriques par rapport à l'axe imaginaire.

L'espace des matrices hamiltoniennes est une algèbre de Lie ${\mathfrak {Sp}}(2n)$ ^[1].

Opérateurs hamiltoniens

Soit V un espace vectoriel, doté d'une forme symplectique $\Omega$ . Une application linéaire $A:\;V\mapsto V$ est appelée opérateur hamiltonien par rapport à $\Omega$ si l'application $x,y\mapsto \Omega (A(x),y)$ est symétrique. De manière équivalente, elle doit satisfaire :

\Omega (A(x),y)=-\Omega (x,A(y))

Soit une base $e_{1},...e_{2n}$ de V telle que $\Omega$ soit écrite $\sum _{i}e_{i}\wedge e_{n+i}$ . un opérateur linéaire est hamiltonien par rapport à $\Omega$ si et seulement si sa matrice dans cette base est hamiltonienne^[2]. Cette définition implique que le carré d'une matrice hamiltonienne est anti-hamiltonien. L'exponentiel d'une matrice hamiltonienne est symplectique, et le logarithme d'une matrice symplectique est hamiltonien.

Voir aussi

Matrice symplectique

Références

↑ (en) Alex J. Dragt, The Symplectic Group and Classical Mechanics'' Annals of the New York Academy of Sciences (2005) 1045 (1), 291-307.
↑ (en) William C. Waterhouse, The structure of alternating-Hamiltonian matrices, Linear Algebra and its Applications, Volume 396, 1^er février 2005, Pages 385-390

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Hamiltonian matrix » (voir la liste des auteurs).

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