The Iwasawa decomposition of G = KAN is given by [4, p.285]

Abstract. We obtain an explicit Iwasawa decomposition of the symplectic matrices, complex or real, in terms of the Cholesky factorization for positive definite n×n matrices. We also provide a MATLAB program to compute the decomposition. 1. Iwasawa decomposition of the symplectic groups Let G be the real (noncompact) symplectic group [3, p.129] (the notation there is Spn), [4, p.265] G: = Spn(R) = {g ∈ SL2n(R) : gTJng = Jn}, Jn