# Pascal matrix

Short description: Infinite matrices with Pascal's triangle as elements

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 × 5 matrices are:

$\displaystyle{ 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}\,\,\, }$$\displaystyle{ U_5 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}\,\,\, }$$\displaystyle{ 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.[1]

## Definition

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

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

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

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

with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... (sequence A006134 in the OEIS).

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

$\displaystyle{ \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 ] ;\quad \\ \\ & 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 ] ; \\ \\ \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} }$

It is important to note that 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.

## 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 × 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials

$\displaystyle{ \begin{array}{lll} & LAG_7=\exp \left ( \left [ \begin{smallmatrix} . & . & . & . & . & . & . \\ 1 & . & . & . & . & . & . \\ . & 4 & . & . & . & . & . \\ . & . & 9 & . & . & . & . \\ . & . & . & 16 & . & . & . \\ . & . & . & . & 25 & . & . \\ . & . & . & . & . & 36 & . \end{smallmatrix} \right ] \right ) = \left [ \begin{smallmatrix} 1 & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & . \\ 2 & 4 & 1 & . & . & . & . \\ 6 & 18 & 9 & 1 & . & . & . \\ 24 & 96 & 72 & 16 & 1 & . & . \\ 120 & 600 & 600 & 200 & 25 & 1 & . \\ 720 & 4320 & 5400 & 2400 & 450 & 36 & 1 \end{smallmatrix} \right ] ;\quad \end{array} }$

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)

$\displaystyle{ \begin{array}{lll} & LAH_7 = \exp \left ( \left [ \begin{smallmatrix} . & . & . & . & . & . & . \\ 2 & . & . & . & . & . & . \\ . & 6 & . & . & . & . & . \\ . & . &12 & . & . & . & . \\ . & . & . & 20 & . & . & . \\ . & . & . & . & 30 & . & . \\ . & . & . & . & . & 42 & . \end{smallmatrix} \right ] \right ) = \left [ \begin{smallmatrix} 1 & . & . & . & . & . & . & . \\ 2 & 1 & . & . & . & . & . & . \\ 6 & 6 & 1 & . & . & . & . & . \\ 24 & 36 & 12 & 1 & . & . & . & . \\ 120 & 240 & 120 & 20 & 1 & . & . & . \\ 720 & 1800 & 1200 & 300 & 30 & 1 & . & . \\ 5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & . \\ 40320 & 141120 & 141120 & 58800 & 11760 & 1176 & 56 & 1 \end{smallmatrix} \right ] ;\quad \end{array} }$

Using v(v − 1) instead provides a diagonal shifting to bottom-right.

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:

$\displaystyle{ \begin{array}{lll} & GS_7 = \exp \left ( \left [ \begin{smallmatrix} . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & . & . & . & . & . & . \\ . & 3 & . & . & . & . & . \\ . & . & 6 & . & . & . & . \\ . & . & . & 10 & . & . & . \\ . & . & . & . & 15 & . & . \end{smallmatrix} \right ] \right ) = \left [ \begin{smallmatrix} 1 & . & . & . & . & . & . \\ . & 1 & . & . & . & . & . \\ 1 & . & 1 & . & . & . & . \\ . & 3 & . & 1 & . & . & . \\ 3 & . & 6 & . & 1 & . & . \\ . & 15 & . & 10 & . & 1 & . \\ 15 & . & 45 & . & 15 & . & 1 \end{smallmatrix} \right ] ;\quad \end{array} }$

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.)

Another variant can be obtained by extending the original matrix to negative values:

$\displaystyle{ \begin{array}{lll} & \exp \left ( \left [ \begin{smallmatrix} . & . & . & . & . & . & . & . & . & . & . & . \\ -5& . & . & . & . & . & . & . & . & . & . & . \\ . &-4 & . & . & . & . & . & . & . & . & . & . \\ . & . &-3 & . & . & . & . & . & . & . & . & . \\ . & . & . &-2 & . & . & . & . & . & . & . & . \\ . & . & . & . &-1 & . & . & . & . & . & . & . \\ . & . & . & . & . & 0 & . & . & . & . & . & . \\ . & . & . & . & . & . & 1 & . & . & . & . & . \\ . & . & . & . & . & . & . & 2 & . & . & . & . \\ . & . & . & . & . & . & . & . & 3 & . & . & . \\ . & . & . & . & . & . & . & . & . & 4 & . & . \\ . & . & . & . & . & . & . & . & . & . & 5 & . \end{smallmatrix} \right ] \right ) = \left [ \begin{smallmatrix} 1 & . & . & . & . & . & . & . & . & . & . & . \\ -5 & 1 & . & . & . & . & . & . & . & . & . & . \\ 10 & -4 & 1 & . & . & . & . & . & . & . & . & . \\ -10 & 6 & -3 & 1 & . & . & . & . & . & . & . & . \\ 5 & -4 & 3 & -2 & 1 & . & . & . & . & . & . & . \\ -1 & 1 & -1 & 1 & -1 & 1 & . & . & . & . & . & . \\ . & . & . & . & . & 0 & 1 & . & . & . & . & . \\ . & . & . & . & . & . & 1 & 1 & . & . & . & . \\ . & . & . & . & . & . & 1 & 2 & 1 & . & . & . \\ . & . & . & . & . & . & 1 & 3 & 3 & 1 & . & . \\ . & . & . & . & . & . & 1 & 4 & 6 & 4 & 1 & . \\ . & . & . & . & . & . & 1 & 5 & 10 & 10 & 5 & 1 \end{smallmatrix} \right ] . \end{array} }$
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