In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are:
There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity.[1]
The non-zero elements of a Pascal matrix are given by the binomial coefficients:
such that the indices i, j start at 0, and ! denotes the factorial.
The matrices have the pleasing relationship Sn = LnUn. From this it is easily seen that all three matrices have determinant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for both Ln and Un. In other words, matrices Sn, Ln, and Un are unimodular, with Ln and Un having trace n.
The trace of Sn is given by
tr(Sn)=
| n | |
| \sum | |
| i=1 |
| [2(i-1)]! | |
| [(i-1)!]2 |
=
| n-1 | |
| \sum | |
| k=0 |
| (2k)! | |
| (k!)2 |
A Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix. The example below constructs a 7 × 7 Pascal matrix, but the method works for any desired n × n Pascal matrices. The dots in the following matrices represent zero elements.
\begin{array}{lll} &L7=\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&2&.&.&.&.&.\\ .&.&3&.&.&.&.\\ .&.&.&4&.&.&.\\ .&.&.&.&5&.&.\\ .&.&.&.&.&6&. \end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&.&.&.&.&.&.\\ 1&1&.&.&.&.&.\\ 1&2&1&.&.&.&.\\ 1&3&3&1&.&.&.\\ 1&4&6&4&1&.&.\\ 1&5&10&10&5&1&.\\ 1&6&15&20&15&6&1\end{smallmatrix} \right] ; \\ \\ &U7=\exp \left(\left[ \begin{smallmatrix} {\color{white}1}.&1&.&.&.&.&.\\ {\color{white}1}.&.&2&.&.&.&.\\ {\color{white}1}.&.&.&3&.&.&.\\ {\color{white}1}.&.&.&.&4&.&.\\ {\color{white}1}.&.&.&.&.&5&.\\ {\color{white}1}.&.&.&.&.&.&6\\ {\color{white}1}.&.&.&.&.&.&.\end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&1&1&1&1&1&1\\ .&1&2&3&4&5&6\\ .&.&1&3&6&10&15\\ .&.&.&1&4&10&20\\ .&.&.&.&1&5&15\\ .&.&.&.&.&1&6\\ .&.&.&.&.&.&1\end{smallmatrix} \right] ; \\ \\ \therefore&S7 =\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&2&.&.&.&.&.\\ .&.&3&.&.&.&.\\ .&.&.&4&.&.&.\\ .&.&.&.&5&.&.\\ .&.&.&.&.&6&. \end{smallmatrix} \right] \right) \exp \left(\left[ \begin{smallmatrix} {\color{white}i}.&1&.&.&.&.&.\\ {\color{white}i}.&.&2&.&.&.&.\\ {\color{white}i}.&.&.&3&.&.&.\\ {\color{white}i}.&.&.&.&4&.&.\\ {\color{white}i}.&.&.&.&.&5&.\\ {\color{white}i}.&.&.&.&.&.&6\\ {\color{white}i}.&.&.&.&.&.&.\end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&1&1&1&1&1&1\\ 1&2&3&4&5&6&7\\ 1&3&6&10&15&21&28\\ 1&4&10&20&35&56&84\\ 1&5&15&35&70&126&210\\ 1&6&21&56&126&252&462\\ 1&7&28&84&210&462&924 \end{smallmatrix} \right]. \end{array}
One cannot simply assume exp(A) exp(B) = exp(A + B), for n × n matrices A and B; this equality is only true when AB = BA (i.e. when the matrices A and B commute). In the construction of symmetric Pascal matrices like that above, the sub- and superdiagonal matrices do not commute, so the (perhaps) tempting simplification involving the addition of the matrices cannot be made.
A useful property of the sub- and superdiagonal matrices used for the construction is that both are nilpotent; that is, when raised to a sufficiently great integer power, they degenerate into the zero matrix. (See shift matrix for further details.) As the n × n generalised shift matrices we are using become zero when raised to power n, when calculating the matrix exponential we need only consider the first n + 1 terms of the infinite series to obtain an exact result.
Interesting variants can be obtained by obvious modification of the matrix-logarithm PL7 and then application of the matrix exponential.
The first example below uses the squares of the values of the log-matrix and constructs a 7 × 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials
\begin{array}{lll} &LAG7=\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&4&.&.&.&.&.\\ .&.&9&.&.&.&.\\ .&.&.&16&.&.&.\\ .&.&.&.&25&.&.\\ .&.&.&.&.&36&. \end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&.&.&.&.&.&.\\ 1&1&.&.&.&.&.\\ 2&4&1&.&.&.&.\ 6&18&9&1&.&.&.\ 24&96&72&16&1&.&.\ 120&600&600&200&25&1&.\ 720&4320&5400&2400&450&36&1\end{smallmatrix} \right] ; \end{array}
The Laguerre-matrix is actually used with some other scaling and/or the scheme of alternating signs. (Literature about generalizations to higher powers is not found yet)
The second example below uses the products v(v + 1) of the values of the log-matrix and constructs a 7 × 7 "Lah"- matrix (or matrix of coefficients of Lah numbers)
\begin{array}{lll} &LAH7=\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.\\ 2&.&.&.&.&.&.\\ .&6&.&.&.&.&.\\ .&.&12&.&.&.&.\\ .&.&.&20&.&.&.\\ .&.&.&.&30&.&.\\ .&.&.&.&.&42&. \end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&.&.&.&.&.&.&.\\ 2&1&.&.&.&.&.&.\\ 6&6&1&.&.&.&.&.\\ 24&36&12&1&.&.&.&.\\ 120&240&120&20&1&.&.&.\\ 720&1800&1200&300&30&1&.&.\\ 5040&15120&12600&4200&630&42&1&.\\ 40320&141120&141120&58800&11760&1176&56&1 \end{smallmatrix} \right] ; \end{array}
The third example below uses the square of the original PL7-matrix, divided by 2, in other words: the first-order binomials (binomial(k, 2)) in the second subdiagonal and constructs a matrix, which occurs in context of the derivatives and integrals of the Gaussian error function:
\begin{array}{lll} &GS7=\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.\\ .&.&.&.&.&.&.\\ 1&.&.&.&.&.&.\\ .&3&.&.&.&.&.\\ .&.&6&.&.&.&.\\ .&.&.&10&.&.&.\\ .&.&.&.&15&.&. \end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&.&.&.&.&.&.\ .&1&.&.&.&.&.\ 1&.&1&.&.&.&.\ .&3&.&1&.&.&.\ 3&.&6&.&1&.&.\ .&15&.&10&.&1&.\ 15&.&45&.&15&.&1\end{smallmatrix} \right] ; \end{array}
Another variant can be obtained by extending the original matrix to negative values:
\begin{array}{lll} &\exp \left(\left[ \begin{smallmatrix} .&.&.&.&.&.&.&.&.&.&.&.\\ -5&.&.&.&.&.&.&.&.&.&.&.\\ .&-4&.&.&.&.&.&.&.&.&.&.\\ .&.&-3&.&.&.&.&.&.&.&.&.\\ .&.&.&-2&.&.&.&.&.&.&.&.\\ .&.&.&.&-1&.&.&.&.&.&.&.\\ .&.&.&.&.&0&.&.&.&.&.&.\\ .&.&.&.&.&.&1&.&.&.&.&.\\ .&.&.&.&.&.&.&2&.&.&.&.\\ .&.&.&.&.&.&.&.&3&.&.&.\\ .&.&.&.&.&.&.&.&.&4&.&.\\ .&.&.&.&.&.&.&.&.&.&5&. \end{smallmatrix} \right] \right) = \left[ \begin{smallmatrix} 1&.&.&.&.&.&.&.&.&.&.&.\\ -5&1&.&.&.&.&.&.&.&.&.&.\\ 10&-4&1&.&.&.&.&.&.&.&.&.\\ -10&6&-3&1&.&.&.&.&.&.&.&.\\ 5&-4&3&-2&1&.&.&.&.&.&.&.\\ -1&1&-1&1&-1&1&.&.&.&.&.&.\\ .&.&.&.&.&0&1&.&.&.&.&.\\ .&.&.&.&.&.&1&1&.&.&.&.\\ .&.&.&.&.&.&1&2&1&.&.&.\\ .&.&.&.&.&.&1&3&3&1&.&.\\ .&.&.&.&.&.&1&4&6&4&1&.\\ .&.&.&.&.&.&1&5&10&10&5&1\end{smallmatrix} \right] . \end{array}