  We analyze the spectral properties of correlation matrices between distinct
statistical systems. Such matrices are intrinsically non symmetric, and lend
themselves to extend the spectral analyses usually performed on standard
Pearson correlation matrices to the realm of complex eigenvalues. We employ
some recent random matrix theory results on the average eigenvalue density of
this type of matrices to distinguish between noise and non trivial correlation
structures, and we focus on financial data as a case study. Namely, we employ
daily prices of stocks belonging to the American and British stock exchanges,
and look for the emergence of correlations between two such markets in the
eigenvalue spectrum of their non symmetric correlation matrix. We find several
non trivial results, also when considering time-lagged correlations over short
lags, and we corroborate our findings by additionally studying the asymmetric
correlation matrix of the principal components of our datasets.
