1. A linear transformation matrix calculating apparatus configured to obtain a linear transformation matrix used by a pattern recognizer for linearly transforming an input subspace and dictionary subspaces as a preprocessing for a calculation of similarities by using Mutual Subspace Method, the pattern recognizer being configured to obtain the input subspace from an input pattern, calculate the similarities between the input subspace and a plurality of the dictionary subspaces which belong to respective categories specified in advance by using the Mutual Subspace Method, and classify the input pattern into one of the categories on the basis of the similarities, comprising: a linear transformation unit configured to linearly transform the plurality of dictionary subspaces which belong to the respective categories respectively by the linear transformation matrix; a loss function calculating unit configured to calculate loss functions using the similarities among the plurality of linearly transformed dictionary subspaces; a differential calculating unit configured to calculate a differential parameter obtained by differentiation of the loss function with respect to the linear transformation matrix; and an updating unit configured to calculate a new linear transformation matrix by the differential parameter and the linear transformation matrix by Steepest Descent Method and update the new linear transformation matrix as the linear transformation matrix used in the linear transformation unit.