Pascal Matrix
المؤلف:
Abbott, P
المصدر:
"Tricks of the Trade: Pascal Matrices." Mathematica J. 9
الجزء والصفحة:
...
9-1-2021
1053
Pascal Matrix
Three types of 
matrices can be 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
The Pascal 
-matrix or order 
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
 |
(4)
|
and
 |
(5)
|
(Edelman and Strang).
Edelman and Strang give four proofs of the identity (5), the most straightforward of which is
where Einstein summation has been used.
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.
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