1. A personal authentication system comprising: a matrix generation unit configured to classify a plurality of feature amounts of face image data, which are recorded in advance, into classes and to calculate as a mapping matrix, coefficients of a linear discriminant equation which uses said recorded plurality of feature amounts as variables; a feature extraction unit configured to extract a first feature amount from first face image data and to extract a second feature amount from second face image data; a feature transformation unit configured to, by using said mapping matrix, transform said first feature amount into a first transformed feature amount and transform said second feature amount into a second transformed feature amount; a processing unit configured to calculate as a similarity, a normalized correlation value between said first transformed feature amount and said second transformed feature amount; and a data matching unit configured to judge that said first face image data and said second face image data are image data of same person when said similarity exceeds a predetermined threshold, wherein said matrix generation unit calculates a first normalized correlation value between a first average vector of feature amounts belonging to a first class and a first feature amount belonging to said first class, said matrix generation unit calculates a second normalized correlation value between a second average vector of feature amounts in a second class and said first feature amount, said second normalized correlation value is a maximum value of normalized correlation values between said first feature amount and average vectors of feature amounts in respective classes other than said first class, said matrix generation unit updates said mapping matrix by using a steepest descent method such that a variation of a loss with respect to said mapping matrix approaches zero, said loss is based on said first normalized correlation value and said second normalized correlation value, and said feature transformation unit transforms feature amounts extracted by said feature extraction unit into transformed feature amounts by using said updated mapping matrix.