The traditional linear discriminant analysis (LDA) is a classical dimensionality reduction method. But there are two problems with LDA. One is the small-sample-size (SSS) problem, and the other is the classification ability of LDA is not very strong. In this paper, by studying several common methods of LDA, an implicit rule of the criterion of LDA is found, and then a general framework of LDA combined with matrix function is presented. Based on this framework, the polynomials are used to reconstruct the LDA criterion, and then a new method called polynomial linear discriminant analysis (PLDA) is proposed. The proposed PLDA method has two contributions: first, it addresses the small-sample-size problem of LDA, and second, it enhances the distance of the between-class sample through polynomial function mapping, improving the classification ability. Experimental results on ORL, FERET, AR, and Coil datasets show the superiority of PLDA over existing variations of LDA. [ABSTRACT FROM AUTHOR]