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The linear algebra of the Pascal matrix. (English) Zbl 0755.15012

Authors’ summary: 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 \(P_ nP^ T_ n\) 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 \(k\)th powers is given.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
65F05 Direct numerical methods for linear systems and matrix inversion
05A19 Combinatorial identities, bijective combinatorics
11D04 Linear Diophantine equations
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References:

[1] Brawer, R., Potenzen der Pascalmatrix and eine Identitat der Kombinatorik, Elem. der Math., 45, 107-110 (1990) · Zbl 0699.05007
[2] Horn, R. A.; Johnson, C. A., Matrix Analysis (1985), Cambridge U.P: Cambridge U.P Cambridge · Zbl 0576.15001
[3] Riordan, J., Combinatorial Identities (1968), Wiley: Wiley New York · Zbl 0194.00502
[4] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0423.65002
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