Stability radii of symplectic and Hamiltonian matrices

We review the Krein-Gel'fand-Lidskii theory of strong stability of Hamiltonian and symplectic matrices in order to nd a quantitative measure of the strong stability. As a result, formulas for the 2-norm distance from a strongly stable Hamiltonian or symplectic matrix to the set of unstable matrices are derived. 1 Deenition and simple properties of symplectic and Hamiltonian matrices Throughout the paper the superscript denotes the hermitian conjugation, and I is the identity matrix. The Krein-Gel'fand-Lidskii theory ((4], Chapter 3), which is main tool in study of stability of symplectic and Hamiltonian matrices, is valid for general G-unitary and G-Hamiltonian matrices. Let G 2 C nn be a nonsingular hermitian matrix (i.e. det G 6 = 0, G = G) which is neither positive nor negative deenite. In applications the matrix G often equals iJ, where i is the imaginary unit and J = 0 ?I k I k 0 !. Together with the standard scalar product (x; y) we deene a new, indeenite scalar product < x; y >= (Gx; y). Note that the number < x; x > is real but unlike (x; x) it need not be positive, hence the term indeenite. Given a matrix A 2 C nn , the matrix A G deened by < Ax; y >=< x; A G y > (x, y are arbitrary vectors in C nn) is said to be G-adjoint to A.