Pascal matrix
Definition The Pascal matrix^{} $P$ of order $n$ is the real square $n\times n$ matrix whose entries are [1]
$${P}_{ij}=\left(\genfrac{}{}{0pt}{}{i+j-2}{j-1}\right).$$ |
For $n=5$,
$$P=\left(\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 10\hfill & \hfill 15\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 10\hfill & \hfill 20\hfill & \hfill 35\hfill \\ \hfill 1\hfill & \hfill 5\hfill & \hfill 15\hfill & \hfill 35\hfill & \hfill 70\hfill \end{array}\right),$$ |
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.
Title | Pascal matrix |
---|---|
Canonical name | PascalMatrix |
Date of creation | 2013-03-22 13:44:54 |
Last modified on | 2013-03-22 13:44:54 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 6 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 65F35 |
Classification | msc 15A12 |
Classification | msc 15A09 |
Classification | msc 15A57 |