Publication date: 05/24/2021

Some multivariate techniques require the calculation of inverse covariance matrices. This section describes how the SVD can be used to calculate the inverse of a covariance matrix.

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)+, where the diagonal elements of Diag(Λ) are replaced by their reciprocals. This defines a generalize inverse of S as follows:

This generalized inverse can be calculated using only the SVD.

For more information about the application of the SVD for wide linear discriminant analysis, see Wide Linear Discriminant Method.

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