The last two decades have witnessed the rapid development of large dimensional matrices in probability and statistics. A lot of efforts have been made to understand the statistical properties of these massive matrices. However, less is known about the algorithmic properties.  In this talk, I will report some recent results towards this direction.  Specifically, we investigate the iteration-wise property of the conjugate gradient algorithm when applied for general spiked covariance matrices. We establish the first and second order asymptotics of the errors and residuals for any iteration up to log n.  Our calculation relies on a Riemann-Hilbert approach and a comprehensive study of the eigenvector empirical spectral distribution in Random Matrix Theory.  

This talk is based on joint works with Thomas Trogdon.