In this paper, we study canonical correlation analysis (CCA), which has become a powerful tool in multivariate data analysis for finding the correlations between two sets of multidimensional variables. The main contributions of the paper are: (i) to reveal the equivalent relationship between a recursive formula and a trace formula for the multiple CCA problem; (ii) to obtain the explicit characterization of all solutions for the multiple CCA problem even the covariance matrices are singular; (iii) to develop a new sparse CCA algorithm; and (iv) to establish the equivalent relationship between the uncorrelated linear discriminant analysis and the CCA problem. We test several simulated and real world data sets in gene classification and cross-language document retrieval to demonstrate the effectiveness of the proposed algorithm. The performance of the proposed method is competitive with the state-of-the-art sparse CCA algorithms.