  We derive the exact form of the eigenvalue spectra of correlation matrices
derived from a set of time-shifted, finite Brownian random walks (time-series).
These matrices can be seen as random, real, asymmetric matrices with a special
structure superimposed due to the time-shift. We demonstrate that the
associated eigenvalue spectrum is circular symmetric in the complex plane for
large matrices. This fact allows us to exactly compute the eigenvalue density
via an inverse Abel-transform of the density of the symmetrized problem. We
demonstrate the validity of this approach by numerically computing eigenvalue
spectra of lagged correlation matrices based on uncorrelated, Gaussian
distributed time-series. We then compare our theoretical findings with
eigenvalue densities obtained from actual high frequency (5 min) data of the
S&P500 and discuss the observed deviations. We identify various non-trivial,
non-random patterns and find asymmetric dependencies associated with
eigenvalues departing strongly from the Gaussian prediction in the imaginary
part. For the same time-series, with the market contribution removed, we
observe strong clustering of stocks, i.e. causal sectors. We finally comment on
the time-stability of the observed patterns.
