A factorization theorem for 4x4 symplectic matrices - Hernandez, Mary Recylee D. (Animo Repository, 2023)
Let A∈GL(n,ℂ). Let S1={β1,β2,...,βn} and S2={γ1,γ2,...,γn} be subsets of ℂ\{0}. We say that A realizes (S1,S2) if there exist B,C∈GL(n,ℂ) such that A=BC with σ(B)=S1 and σ(C)=S2. If both B,C are from a subgroup G≤GL(n,ℂ), we say that (S1,S2) is realized by A in G. Sourour showed that if S1={β1,β2,...,βn} and S2={γ1,γ2,...,γn}⊆𝔽\{0} are given, a nonscalar A∈GL(n,𝔽) realizes (S1,S2) if and only if det A=Π_(j=1)^n βjγj. We take G to be the 2nx2n symplectic group Sp(2n,ℂ) and determine if there exists pair of sets (S1,S2) which are realizable by a matrix A∈Sp(4,ℂ).