A matrix S โ C 2mร2m is symplectic if SJ S * = J , where
Symplectic matrices play an important role in the analysis and numerical solution of matrix problems involving the indefinite inner product x * (iJ )y. In this paper we provide several matrix factorizations related to symplectic matrices. We introduce a singular value-like decomposition B = QDS -1 for any real matrix B โ R nร2m , where Q is real orthogonal, S is real symplectic, and D is permuted diagonal. We show the relation between this decomposition and the canonical form of real skew-symmetric matrices and a class of Hamiltonian matrices. We also show that if S is symplectic it has the structured singular value decomposition S = UDV * , where U, V are unitary and symplectic, D = diag( , -1 ) and is positive diagonal. We study the BJ B T factorization of real skew-symmetric matrices. The BJ B T factorization has the applications in solving the skew-symmetric systems of linear equations, and the eigenvalue problem for skew-symmetric/symmetric pencils. The BJ B T factorization is not unique, and in numerical application one requires the factor B with small norm and condition number to improve the numerical stability. By employing the singular value-like decomposition and the singular value decomposition of symplectic matrices we give the general formula for B with minimal norm and condition number.