  The present paper uses spectral theory of linear operators to construct
approximately minimal realizations of weighted languages. Our new contributions
are: (i) a new algorithm for the SVD decomposition of infinite Hankel matrices
based on their representation in terms of weighted automata, (ii) a new
canonical form for weighted automata arising from the SVD of its corresponding
Hankel matrix and (iii) an algorithm to construct approximate minimizations of
given weighted automata by truncating the canonical form. We give bounds on the
quality of our approximation.
