A Note on the Factorization of a Matrix Inverse

Abstract

The decomposition of a matrix as a product of a lower triangular with ones on the diagonal and an upper triangular matrix is useful for solving systems of linear equations. For a given non singular matrix, this type of decomposition is unique and algorithms exist to obtain the two factors. However, in certain problems the factorization of the inverse matrix may be of interest. This note presents an algorithm for factoring the inverse matrix using simple operations of elements from the original matrix. As examples, we give factorizations for several well-known and widely used correlation matrices. The usefulness and practicality of these factorizations are provided in an application of statistical modeling using unbiased estimating equations.

Keywords:

• Cholesky decomposition
• Estimating equations

Mathematics Subject Classification:

• 15A23
• 15A09
• 62J02
• 62J12

Acknowledgments

The author is grateful to the Editor, the Associate Editor, and the referee for their constructive comments which improved the presentation.