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Pascal Matrix


Three types of n×n matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix S_n with (S)_(ij)=(i+j; i), a lower triangular matrix L_n with (L)_(ij)=(i; j), and an upper triangular matrix U_n with (U)_(ij)=(j; i), where i,j=0, 1, ..., n-1. For example, for n=4, these would be given by

S_4=[1 1 1 1; 1 2 3 4; 1 3 6 10; 1 4 10 20]
(1)
L_4=[1   ; 1 1  ; 1 2 1 ; 1 3 3 1]
(2)
U_4=[1 1 1 1;  1 2 3;   1 3;    1].
(3)

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

These matrices have some amazing properties. In particular, their determinants are all equal to 1

 |S_n|=|L_n|=|U_n|=1,
(4)

and

 S_n=L_nU_n
(5)

(Edelman and Strang).

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

(LU)_(ij)=L_(ik)U_(kj)
(6)
=sum_(k=0)^(n-1)(i; k)(j; k)
(7)
=(i+j; i)
(8)
=(S)_(ij),
(9)

where Einstein summation has been used.


See also

Pascal's Triangle

Explore with Wolfram|Alpha

References

Abbott, P. "Tricks of the Trade: Pascal Matrices." Mathematica J. 9, 691-694, 2005.Edelman, A. and Strang, G. "Pascal Matrices." http://web.mit.edu/18.06/www/pascal-work.pdf.Strang, G. Introduction to Linear Algebra, 3rd ed. Wellesley-Cambridge Press, 2003.

Referenced on Wolfram|Alpha

Pascal Matrix

Cite this as:

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

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