We present a constructive existence proof that every real skew-Hamiltonian
matrix W has a real Hamiltonian square root. The key step in this construct
ion shows how one may bring any such W into a real quasi-Jordan canonical f
orm via symplectic similarity. We show further that every W has infinitely
many real Hamiltonian square roots, and give a lower bound on the dimension
of the set of all such square roots. Some extensions to complex matrices a
re also presented. (C) 1999 Elsevier Science Inc. All rights reserved.