Pascal matrix

In mathematics, particularly 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&thinsp;×&thinsp;5 matrices are:

$$L_5 = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 1 & 3 & 3 & 1 & 0 \\ 1 & 4 & 6 & 4 & 1 \end{pmatrix}\,\,\,$$$$U_5 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}\,\,\,$$$$S_5 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 70 \end{pmatrix}=L_5 \times U_5$$ There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity.

Definition
The non-zero elements of a Pascal matrix are given by the binomial coefficients:

$$L_{ij} = {i \choose j} = \frac{i!}{j!(i-j)!}, j \le i$$ $$U_{ij} = {j \choose i} = \frac{j!}{i!(j-i)!}, i \le j$$ $$S_{ij} = {i+j \choose i} = {i+j \choose j} = \frac{(i+j)!}{i!j!}$$

such that the indices i, j start at 0, and ! denotes the factorial.

Properties
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
 * $$\text{tr}(S_n) = \sum^n_{i=1} \frac{ [ 2(i-1) ] !}{[(i-1)!]^2} = \sum^{n-1}_{k=0} \frac{ (2k) !}{(k!)^2}$$

with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... .

Construction
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&thinsp;×&thinsp;7 Pascal matrix, but the method works for any desired n&thinsp;×&thinsp;n Pascal matrices. The dots in the following matrices represent zero elements.



\begin{array}{lll} & L_7=\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 ] \\ \\ & U_7=\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 ] \\ \\
 * \quad

\therefore & S_7 =\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)&thinsp;exp(B) = exp(A + B), for n&thinsp;×&thinsp;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&thinsp;×&thinsp;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 +&thinsp;1 terms of the infinite series to obtain an exact result.

Variants
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&thinsp;×&thinsp;7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials

\begin{array}{lll} & LAG_7=\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} $$
 * \quad

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 +&thinsp;1) of the values of the log-matrix and constructs a 7&thinsp;×&thinsp;7 "Lah"- matrix (or matrix of coefficients of Lah numbers)


 * $$\begin{array}{lll}

& LAH_7 = \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} $$ Using v(v &minus; 1) instead provides a diagonal shifting to bottom-right.
 * \quad

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} & GS_7 = \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} $$ 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.)
 * \quad

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} $$