Computing the Iwasawa decomposition of a symplectic matrix by Cholesky factorization

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) : g Jng = Jn}, Jn = ( 0 In −In 0 ) . For example,   cosh t sinh t 0 sinh t sinh t cosh t sinh t 0 0 0 cosh t − sinh t 0 0 − sinh t cosh t   ∈ Sp4(R), t ∈ R. By block multiplication, the elements of G are of the form [3, p.128] (1.1) ( A B C D ) , where A C = C A, B D = D B, A D − C B = In. The Iwasawa decomposition of G = KAN is given by [4, p.285] K = {( A B −B A ) : A + iB ∈ U(n) } = O(2n) ∩ Spn(R), A = {diag (a1, . . . , an, a−1 1 , . . . , a−1 n ) : a1, . . . , an > 0}, N = {(A B 0 (A−1)T ) : A real unit upper triangular, AB = BA } . That is, if g ∈ G, then g can be written in the form g = kan, k ∈ K, a ∈ A, n ∈ N , and the decomposition is unique. When g ∈ G is viewed as a 2n× 2n real nonsingular matrix, we have the usual QR decomposition [3, p.143] of g = k′a′n′, where k′ is special orthogonal, a′ is positive diagonal and n′ is real unit upper triangular and the decomposition is unique. However, it is not the Iwasawa decomposition of g = kan ∈ G, k ∈ K, The results in the paper were presented in R.C. Thompson’s Matrix Meeting held at San Jose State University, Nov., 2004 2000 Mathematics Subject Classification. Primary 15A23, 22E46 c ©0000 (copyright holder)