Denote the standardized data matrix by Xs and define S = Xs’Xs. The singular value decomposition allows you to write S as follows:

If S is of full rank, then V is a p by p orthonormal matrix, and you can write S-1 as follows:

If S is not of full rank, then Diag(Λ)-1 can be replaced with a generalized inverse, Diag(Λ)-1, where the diagonal elements of Diag(Λ) are replaced by their reciprocals. This defines a generalize inverse of S as follows:

To see the specific details behind the application of the SVD for wide linear discriminant analysis, see Wide Linear Discriminant Method in Discriminant Analysis.