Generalization of a statistical matrix and its factorization

Abstract

We consider a special matrix with $q-$integer coefficients and obtain an LU factorization for its member by giving explicit closed-form formulae of the entries of L and U. Our result is applied to give the closed-form formula of the inverse of the considered matrix. We give the relation between the defined matrix and Helmert matrix which has been used for proving the statistical independence of a number of statistics. Also we find the condition numbers of some matrices for some special values of q.

Keywords:

• Factorization of matrices
• matrix inversion
• $q-$calculus
• matrix norms and conditioning

Mathematics Subject Classification:

• 15A23
• 15A09
• 05A30
• 65F35

Acknowledgments

The authors would like to thank the referees for their valuable comments which helped to improve the paper.