Pascal matrix

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

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

For $n=5$ ,

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.