  We present a brief overview of random matrix theory (RMT) with the objectives
of highlighting the computational results and applications in financial markets
as complex systems. An oft-encountered problem in computational finance is the
choice of an appropriate epoch over which the empirical cross-correlation
return matrix is computed. A long epoch would smoothen the fluctuations in the
return time series and suffers from non-stationarity, whereas a short epoch
results in noisy fluctuations in the return time series and the correlation
matrices turn out to be highly singular. An effective method to tackle this
issue is the use of the power mapping, where a non-linear distortion is applied
to a short epoch correlation matrix. The value of distortion parameter controls
the noise-suppression. The distortion also removes the degeneracy of zero
eigenvalues. Depending on the correlation structures, interesting properties of
the eigenvalue spectra are found. We simulate different correlated Wishart
matrices to compare the results with empirical return matrices computed using
the S&P 500 (USA) market data for the period 1985-2016. We also briefly review
two recent applications of RMT in financial stock markets: (i) Identification
of "market states" and long-term precursor to a critical state; (ii)
Characterization of catastrophic instabilities (market crashes).
