Fisher Discriminant Analysis (FDA) has been widely used as a dimensionality reduction technique. Its application varies from face recognition to speaker recognition. In the past two decades, there have been many variations on the formulation of FDA. Different variations adopt different ways to combine the between-class scatter matrix and the within-class scatter matrix, which are two basic components in FDA. In this paper, we propose the Generalized Fisher Discriminant Analysis (GFDA), which provides a general formulation for FDA. GFDA generalizes the standard FDA as well as many different variants of FDA, such as Regularized Linear Discriminant Analysis (R-LDA), Regularized Kernel Discriminant Analysis (R-KDA), Inverse Fisher Discriminant Analysis (IFDA), and Regularized Fisher Discriminant Analysis (RFDA). GFDA can also degenerate to Principal Component Analysis (PCA). Four special types of GFDA are then applied as dimensionality reduction techniques for speaker recognition, in order to investigate the performance of different variants of FDA. Basically, GFDA provides a convenient way to compare different variants of FDA by simply changing some parameters. It makes it easier to explore the roles that the between-class scatter matrix and the within-class scatter matrix play.