On the conditioning of factors in the SR decomposition

Almost every nonsingular matrix A∈R2m,2mA∈R2m,2m can be decomposed into the product of a symplectic matrix S and an upper J-triangular matrix R. This decomposition is not unique. In this paper we analyze the freedom of choice in the symplectic and the upper J-triangular factors and review several existing suggestions on how to choose the free parameters in the SR decomposition. In particular we consider two choices leading to the minimization of the condition number of the diagonal blocks in the upper J-triangular factor and to the minimization of the conditioning of the corresponding blocks in the symplectic factor. We develop bounds for the extremal singular values of the whole upper J-triangular factor and the whole symplectic factor in terms of the spectral properties of even-dimensioned principal submatrices of the skew-symmetric matrix associated with the SR decomposition. The theoretical results are illustrated on two small examples.