65F35-PascalMatrix.tex

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\pmtitle{Pascal matrix}
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{\bf Definition} The \emph{Pascal matrix} $P$ of order $n$ is the real 
square $n\times n$ matrix whose entries are \cite{higham}
 $$ P_{ij} = { i+j-2 \choose j-1}. $$
 
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 \cite{higham}.
The characteristic polynomial of a Pascal triangle is a reciprocal 
polynomial \cite{higham}.

\begin{thebibliography}{9}
 \bibitem{higham} N.J. Higham, \emph{Accuracy and Stability of Numerical Algorithms},
 2nd ed., SIAM, 2002.
 \end{thebibliography}
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