  The null linear discriminant analysis method is a competitive approach for
dimensionality reduction. The implementation of this method, however, is
computationally expensive. Recently, a fast implementation of null linear
discriminant analysis method using random matrix multiplication with scatter
matrices was proposed. However, if the random matrix is chosen arbitrarily, the
orientation matrix may be rank deficient, and some useful discriminant
information will be lost. In this paper, we investigate how to choose the
random matrix properly, such that the two criteria of the null LDA method are
satisfied theoretically. We give a necessary and sufficient condition to
guarantee full column rank of the orientation matrix. Moreover, the geometric
characterization of the condition is also described.
