On The Products of Commutators of Real J-Symmetries

Authors

• R. J. De La Cruz, Ateneo de Manila University
• Kerish S. Villegas, Ateneo de Manila University

Document Type

Article

Publication Date

2025

Abstract

LetJ2n=[0In−In0]. A 2n-by-2ncomplex matrixAis said to besymplecticifATJA=J. IfAis symplecticand rank(A−I) = 1, thenAis called aJ-symmetry. It is known that every 2n-by-2ncomplex symplectic matrix can be writtenas a product ofn+ 1 commutators ofJ-symmetries. We consider the real case and study the properties of realJ-symmetriesand commutators of realJ-symmetries. We prove that ifAis a 2n-by-2nreal symplectic matrix, with rank(A−I) =m, thenAis a product of3m2−2bm4ccommutators of realJ-symmetries ifJ(A−I) is skew-symmetric, andAis a product of 3dm2ecommutators of realJ-symmetries ifJ(A−I) is not skew-symmetric.

Recommended Citation

De La Cruz, R. J., & Villegas, K. (2025). On the Products of Commutators of Real J-symmetries. The Electronic Journal of Linear Algebra, 41, 193–212.