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 ÃÂ 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.
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 S<sub>n</sub> = L<sub>n</sub>U<sub>n</sub>. 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 L<sub>n</sub> and U<sub>n</sub>. In other words, matrices S<sub>n</sub>, L<sub>n</sub>, and U<sub>n</sub> are unimodular, with L<sub>n</sub> and U<sub>n</sub> having trace n.
The trace of S<sub>n</sub> is given by
with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... .
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 ÃÂ 7 Pascal matrix, but the method works for any desired n ÃÂ n Pascal matrices. The dots in the following matrices represent zero elements.
One cannot simply assume exp(A) exp(B) = exp(A + B), for n ÃÂ 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 ÃÂ 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 PL<sub>7</sub> 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 ÃÂ 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials
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 ÃÂ 7 "Lah"- matrix (or matrix of coefficients of Lah numbers)
Using v(v − 1) instead provides a diagonal shifting to bottom-right.
The third example below uses the square of the original PL<sub>7</sub>-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:
If this matrix is inverted (using, for instance, the negative matrix-logarithm), then this matrix has alternating signs and gives the coefficients of the derivatives (and by extension the integrals) of Gauss' error-function. (Literature about generalizations to greater powers is not found yet.)
Another variant can be obtained by extending the original matrix to negative values: