| symmetric.pascal.matrix {matrixcalc} | R Documentation |
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 |
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.
} {\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}}.
Value
An order n matrix.
Note
If the argument n is not a positive integer, the function presents an error message and stops.
Author(s)
Frederick Novomestky fnovomes@poly.edu
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
S <- symmetric.pascal.matrix( 4 )
print( S )