Pascal matrix

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 ×ばつ 5 matrices are:

$${\displaystyle L_{5}={\begin{pmatrix}1&0&0&0&0\1円&1&0&0&0\1円&2&1&0&0\1円&3&3&1&0\1円&4&6&4&1\end{pmatrix}},円,円,円}$${\displaystyle L_{5}={\begin{pmatrix}1&0&0&0&0\1円&1&0&0&0\1円&2&1&0&0\1円&3&3&1&0\1円&4&6&4&1\end{pmatrix}},円,円,円}$${\displaystyle U_{5}={\begin{pmatrix}1&1&1&1&1\0円&1&2&3&4\0円&0&1&3&6\0円&0&0&1&4\0円&0&0&0&1\end{pmatrix}},円,円,円}$${\displaystyle U_{5}={\begin{pmatrix}1&1&1&1&1\0円&1&2&3&4\0円&0&1&3&6\0円&0&0&1&4\0円&0&0&0&1\end{pmatrix}},円,円,円}$${\displaystyle S_{5}={\begin{pmatrix}1&1&1&1&1\1円&2&3&4&5\1円&3&6&10&15\1円&4&10&20&35\1円&5&15&35&70\end{pmatrix}}=L_{5}\times U_{5}}$${\displaystyle S_{5}={\begin{pmatrix}1&1&1&1&1\1円&2&3&4&5\1円&3&6&10&15\1円&4&10&20&35\1円&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.

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

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

The matrices have the pleasing relationship S_n = L_nU_n. 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_n and U_n. In other words, matrices S_n, L_n, and U_n are unimodular, with L_n and U_n having trace n.

The trace of S_n is given by

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

A Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix. The example below constructs a ×ばつ 7 Pascal matrix, but the method works for any desired n×ばつ n Pascal matrices. The dots in the following matrices represent zero elements.

${\displaystyle {\begin{array}{lll}&L_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&2&.&.&.&.&.\\.&.&3&.&.&.&.\\.&.&.&4&.&.&.\\.&.&.&.&5&.&.\\.&.&.&.&.&6&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\1円&1&.&.&.&.&.\1円&2&1&.&.&.&.\1円&3&3&1&.&.&.\1円&4&6&4&1&.&.\1円&5&10&10&5&1&.\1円&6&15&20&15&6&1\end{smallmatrix}}\right];\quad \\\\&U_{7}=\exp \left(\left[{\begin{smallmatrix}{\color {white}1}.&1&.&.&.&.&.\\{\color {white}1}.&.&2&.&.&.&.\\{\color {white}1}.&.&.&3&.&.&.\\{\color {white}1}.&.&.&.&4&.&.\\{\color {white}1}.&.&.&.&.&5&.\\{\color {white}1}.&.&.&.&.&.&6\\{\color {white}1}.&.&.&.&.&.&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&1&1&1&1&1&1\\.&1&2&3&4&5&6\\.&.&1&3&6&10&15\\.&.&.&1&4&10&20\\.&.&.&.&1&5&15\\.&.&.&.&.&1&6\\.&.&.&.&.&.&1\end{smallmatrix}}\right];\\\\\therefore &S_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&2&.&.&.&.&.\\.&.&3&.&.&.&.\\.&.&.&4&.&.&.\\.&.&.&.&5&.&.\\.&.&.&.&.&6&.\end{smallmatrix}}\right]\right)\exp \left(\left[{\begin{smallmatrix}{\color {white}i}.&1&.&.&.&.&.\\{\color {white}i}.&.&2&.&.&.&.\\{\color {white}i}.&.&.&3&.&.&.\\{\color {white}i}.&.&.&.&4&.&.\\{\color {white}i}.&.&.&.&.&5&.\\{\color {white}i}.&.&.&.&.&.&6\\{\color {white}i}.&.&.&.&.&.&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&1&1&1&1&1&1\1円&2&3&4&5&6&7\1円&3&6&10&15&21&28\1円&4&10&20&35&56&84\1円&5&15&35&70&126&210\1円&6&21&56&126&252&462\1円&7&28&84&210&462&924\end{smallmatrix}}\right].\end{array}}}${\displaystyle {\begin{array}{lll}&L_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&2&.&.&.&.&.\\.&.&3&.&.&.&.\\.&.&.&4&.&.&.\\.&.&.&.&5&.&.\\.&.&.&.&.&6&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\1円&1&.&.&.&.&.\1円&2&1&.&.&.&.\1円&3&3&1&.&.&.\1円&4&6&4&1&.&.\1円&5&10&10&5&1&.\1円&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円&.&.&.&.&.&.\\.&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円&2&3&4&5&6&7\1円&3&6&10&15&21&28\1円&4&10&20&35&56&84\1円&5&15&35&70&126&210\1円&6&21&56&126&252&462\1円&7&28&84&210&462&924\end{smallmatrix}}\right].\end{array}}}

One cannot simply assume exp(A) exp(B) = exp(A + B), for n×ばつ 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×ばつ 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 ×ばつ 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials

${\displaystyle {\begin{array}{lll}&LAG_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&4&.&.&.&.&.\\.&.&9&.&.&.&.\\.&.&.&16&.&.&.\\.&.&.&.&25&.&.\\.&.&.&.&.&36&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\1円&1&.&.&.&.&.\2円&4&1&.&.&.&.\6円&18&9&1&.&.&.\24円&96&72&16&1&.&.\120円&600&600&200&25&1&.\720円&4320&5400&2400&450&36&1\end{smallmatrix}}\right];\quad \end{array}}}${\displaystyle {\begin{array}{lll}&LAG_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&4&.&.&.&.&.\\.&.&9&.&.&.&.\\.&.&.&16&.&.&.\\.&.&.&.&25&.&.\\.&.&.&.&.&36&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\1円&1&.&.&.&.&.\2円&4&1&.&.&.&.\6円&18&9&1&.&.&.\24円&96&72&16&1&.&.\120円&600&600&200&25&1&.\720円&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 ×ばつ 7 "Lah"- matrix (or matrix of coefficients of Lah numbers)

${\displaystyle {\begin{array}{lll}&LAH_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\2円&.&.&.&.&.&.\\.&6&.&.&.&.&.\\.&.&12&.&.&.&.\\.&.&.&20&.&.&.\\.&.&.&.&30&.&.\\.&.&.&.&.&42&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.&.\2円&1&.&.&.&.&.&.\6円&6&1&.&.&.&.&.\24円&36&12&1&.&.&.&.\120円&240&120&20&1&.&.&.\720円&1800&1200&300&30&1&.&.\5040円&15120&12600&4200&630&42&1&.\40320円&141120&141120&58800&11760&1176&56&1\end{smallmatrix}}\right];\quad \end{array}}}${\displaystyle {\begin{array}{lll}&LAH_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\2円&.&.&.&.&.&.\\.&6&.&.&.&.&.\\.&.&12&.&.&.&.\\.&.&.&20&.&.&.\\.&.&.&.&30&.&.\\.&.&.&.&.&42&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.&.\2円&1&.&.&.&.&.&.\6円&6&1&.&.&.&.&.\24円&36&12&1&.&.&.&.\120円&240&120&20&1&.&.&.\720円&1800&1200&300&30&1&.&.\5040円&15120&12600&4200&630&42&1&.\40320円&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:

${\displaystyle {\begin{array}{lll}&GS_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\\.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&3&.&.&.&.&.\\.&.&6&.&.&.&.\\.&.&.&10&.&.&.\\.&.&.&.&15&.&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\\.&1&.&.&.&.&.\1円&.&1&.&.&.&.\\.&3&.&1&.&.&.\3円&.&6&.&1&.&.\\.&15&.&10&.&1&.\15円&.&45&.&15&.&1\end{smallmatrix}}\right];\quad \end{array}}}${\displaystyle {\begin{array}{lll}&GS_{7}=\exp \left(\left[{\begin{smallmatrix}.&.&.&.&.&.&.\\.&.&.&.&.&.&.\1円&.&.&.&.&.&.\\.&3&.&.&.&.&.\\.&.&6&.&.&.&.\\.&.&.&10&.&.&.\\.&.&.&.&15&.&.\end{smallmatrix}}\right]\right)=\left[{\begin{smallmatrix}1&.&.&.&.&.&.\\.&1&.&.&.&.&.\1円&.&1&.&.&.&.\\.&3&.&1&.&.&.\3円&.&6&.&1&.&.\\.&15&.&10&.&1&.\15円&.&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:

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

• Pascal's triangle
• LU decomposition
• Riordan Array

• G. Helms Pascalmatrix in a project of compilation of facts about Numbertheoretical matrices
• G. Helms Gauss-matrix
• Weisstein, Eric W. Gaussian-function
• Weisstein, Eric W. Erf-function
• Weisstein, Eric W. "Hermite Polynomial". Hermite-polynomials
• OEIS sequence A066325 (Coefficients of unitary Hermite polynomials He_n(x)) (Related to Gauss-matrix).

• Matrices (mathematics)
• Triangles of numbers

• Articles with short description
• Short description matches Wikidata
• Articles lacking in-text citations from October 2025
• All articles lacking in-text citations
• Use American English from January 2019
• All Wikipedia articles written in American English