Download Wolfram Notebook

A Pascal matrix is any of three types of matrices obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix with , a lower triangular matrix with , and an upper triangular matrix with , where , 1, ..., . For example, for , these would be given by

All three are integer matrices and hence rational matrices.

The Pascal -matrix of order is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].

Their determinants satisfy

and

(Edelman and Strang 2004).

Edelman and Strang (2004) give four proofs of the identity (5), the most straightforward of which is

where Einstein summation has been used.

Trott (2006) considers the right-justified Pascal-triangle matrix obtained by reversing the columns of ,

For example,

Its eigenvalues are

where , 1, ..., and is the golden ratio (Callan 2000, Callan and Prodinger 2003). These eigenvalues are distinct, so is a diagonalizable matrix. For , they are , , , and . The triangular matrices and each have only the eigenvalue 1.

The infinite lower triangular Pascal matrix is the Riordan array (Shapiro et al. 1991).

See also

Golden Ratio, Integer Matrix, Pascal's Triangle, Rational Matrix

Explore with Wolfram|Alpha

References

Abbott, P. "Tricks of the Trade: Pascal Matrices." Mathematica J. 9, 691-694, 2005.Callan, D. "The Eigenvectors of the Right-Justified Pascal Triangle." 13 Nov 2000. https://arxiv.org/abs/math/0011081.Callan, D. and Prodinger, H. "An Involutory Matrix of Eigenvectors." Fibonacci Quart. 41, 105-107, 2003. https://www.fq.math.ca/Scanned/41-2/callan.pdf.Edelman, A. and Strang, G. "Pascal Matrices." Amer. Math. Monthly 111, 189-197, 2004. https://doi.org/10.1080/00029890.2004.11920065.Melham, R. S. and Cooper, C. "The Eigenvectors of a Certain Matrix of Binomial Coefficients." Fibonacci Quart. 38, 123-126, 2000. https://www.fq.math.ca/Scanned/38-2/melham.pdf.Peele, R. and Stănică, P. "Matrix Powers of Column-Justified Pascal Triangles and Fibonacci Sequences." Fibonacci Quart. 40, 146-152, 2002. https://www.fq.math.ca/Scanned/40-2/peele.pdf.Shapiro, L. W.; Getu, S.; Woan, W.-J.; and Woodson, L. C. "The Riordan Group." Disc. Appl. Math. 34, 229-239, 1991. https://doi.org/10.1016/0166-218X(91)90088-E.Strang, G. Introduction to Linear Algebra, 3rd ed. Wellesley-Cambridge Press, 2003.Trott, M. "Operations on Rational Functions." §1.3 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Pascal Matrix

Cite this as:

Weisstein, Eric W. "Pascal Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PascalMatrix.html

Subject classifications