matrixcalc (version 1.0-3)

symmetric.pascal.matrix: Symmetric Pascal matrix

Description

This function returns an n by n symmetric Pascal matrix.

Usage

symmetric.pascal.matrix(n)

Arguments

n

Order of the matrix

Value

An order \(n\) matrix.

Details

In mathematics, particularly matrix theory and combinatorics, the symmetric Pascal matrix is a square matrix from which you can derive binomial coefficients. The matrix is an order \(n\) symmetric matrix with typical element given by \({S_{i,j}} = {{n!} \mathord{\left/ {\vphantom {{n!} {\left[ {r!\;\left( {n - r} \right)!} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {r!\;\left( {n - r} \right)!} \right]}}\) where \(n = i + j - 2\) and \(r = i - 1\). The binomial coefficients are elegantly recovered from the symmetric Pascal matrix by performing an \(LU\) decomposition as \({\bf{S}} = {\bf{L}}\;{\bf{U}}\).

References

Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, American Mathematical Monthly, April 1993, 100, 372-376.

Edelman, A. and G. Strang, (2004). Pascal Matrices, American Mathematical Monthly, 111(3), 361-385.

Examples

# NOT RUN {
S <- symmetric.pascal.matrix( 4 )
print( S )
# }