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Pascal matrix (Definition)

Definition The Pascal matrix $ P$ of order $ n$ is the real square $ n\times n$ matrix whose entries are [1]

$\displaystyle P_{ij} = { i+j-2 \choose j-1}. $

For $ n=5$,

$\displaystyle P= \begin{pmatrix} 1 & 1 & 1 & 1 & 1\ 1 & 2 & 3 & 4 & 5 \ 1 & 3 & 6 & 10 & 15\ 1 & 4 & 10 & 20 & 35 \ 1 & 5 & 15 & 35 & 70 \end{pmatrix}, $
so we see that the Pascal matrix contains the Pascal triangle on its antidiagonals.

Pascal matrices are ill-conditioned. However, the inverse of the $ n\times n$ Pascal matrix is known explicitly and given in [1]. The characteristic polynomial of a Pascal triangle is a reciprocal polynomial [1].

References

1
N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002.



"Pascal matrix" is owned by bbukh. [ full author list (2) | owner history (1) ]
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Cross-references: reciprocal polynomial, characteristic polynomial, inverse, ill-conditioned, Pascal triangle, contains, matrix, square, real, order
There are 2 references to this entry.

This is version 3 of Pascal matrix, born on 2003-07-14, modified 2004-01-25.
Object id is 4445, canonical name is PascalMatrix.
Accessed 4644 times total.

Classification:
AMS MSC: 65F35 (Numerical analysis :: Numerical linear algebra :: Matrix norms, conditioning, scaling)
 15A12 (Linear and multilinear algebra; matrix theory :: Conditioning of matrices)
 15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )
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