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What are the eigenvalues of a typical matrix with a given set of singular
values? We make the question
precise by considering a random matrix of the form A=UDV, where D is a
diagonal matrix and U,V are independent
unitary matrices sampled from Haar measure.Under certain assumptions on
the
distribution of D, we show that A has a
limiting spectral distribution, and characterize its properties. In
particular, it has the surprising feature that
the support of the limit spectral distribution is a connected annulus.
This
is joint work with Alice Guionnet and
Ofer Zeitouni.
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