The linear algebra of the pascal matrix

Abstract

Pascal's triangle can be represented as a square matrix in two basically different ways: as a lower triangular matrix P n or as a full, symmetric matrix Q n. It has been found that the P n P n T is the Cholesky factorization of Q n . P n can be factorized by special summation matrices. It can be shown that the inverses of these matrices are the operators which perform the Gaussian elimination steps for calculating Cholesky's factorization. By applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems. Finally, an explicit formula for the sum of the kth powers is given.