Markowitzs portfolio selection problem chooses weights for stocks in a portfolio based on a historical covariance matrix of stock returns. Our study proposes to reduce noise in the covariance matrix using a Tikhonov filter function. In addition, we propose a new strategy to resolve the rank deficiency of the covariance matrix, and a method to choose a Tikhonov parameter which determines a filtering intensity. We put the previous estimators into a common framework and compare their filtering functions for eigenvalues of the correlation matrix. Experiments using the daily return data of the most frequently traded stocks in NYSE, AMEX, and NASDAQ show that Tikhonov filtering estimates the covariance matrix better than methods of Sharpe who applies a market-index model, Ledoit et al. who shrink the sample covariance matrix to the market-index covariance matrix, Plerou et al., who suggest truncating the smallest eigenvalues, and Bengtsson et al., who decrease small eigenvalues at a single rate.