Inmatrix theory andcombinatorics, aPascal matrix is amatrix (possiblyinfinite) containing thebinomial coefficients as its elements. It is thus an encoding ofPascal's triangle in matrix form. There are three natural ways to achieve this: as alower-triangular matrix, anupper-triangular matrix, or asymmetric 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.

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

$${\displaystyle L_{ij}=(\genfrac{}{}{0ex}{}{i}{j})=\frac{i!}{j!(i-j)!},j\le i}$$$${\displaystyle U_{ij}=(\genfrac{}{}{0ex}{}{j}{i})=\frac{j!}{i!(j-i)!},i\le j}$$$${\displaystyle S_{ij}=(\genfrac{}{}{0ex}{}{i+j}{i})=(\genfrac{}{}{0ex}{}{i+j}{j})=\frac{(i+j)!}{i!j!}}$$

such that the indicesi,j start at 0, and ! denotes thefactorial.

The matrices have the pleasing relationshipS_n =L_nU_n. From this it is easily seen that all three matrices havedeterminant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for bothL_n andU_n. In other words, matricesS_n,L_n, andU_n areunimodular, withL_n andU_n havingtracen.

The trace ofS_n is given by

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

with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... (sequenceA006134 in theOEIS).

A Pascal matrix can actually be constructed by taking thematrix exponential of a specialsubdiagonal orsuperdiagonal matrix. The example below constructs a 7 × 7 Pascal matrix, but the method works for any desiredn × n Pascal matrices. The dots in the following matrices represent zero elements.

One cannot simply assume exp(A) exp(B) = exp(A + B), forn × n matricesA andB; this equality is only true whenAB =BA (i.e. when the matricesA andBcommute). 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 arenilpotent; that is, when raised to a sufficiently greatinteger power, they degenerate into thezero matrix. (Seeshift matrix for further details.) As then × n generalised shift matrices we are using become zero when raised to powern, when calculating the matrix exponential we need only consider the firstn + 1 terms of theinfinite 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 ofLaguerre 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 productsv(v + 1) of the values of the log-matrix and constructs a 7 × 7 "Lah"- matrix (or matrix of coefficients ofLah numbers)

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

The third example below uses the square of the originalPL₇-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 thederivatives andintegrals of the Gaussianerror function:

If this matrix isinverted (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 tonegative values:

• Pascal's triangle
• LU decomposition
• Riordan Array

• G. HelmsPascalmatrix in a project of compilation of facts aboutNumbertheoretical matrices
• G. HelmsGauss-matrix
• Weisstein, Eric W.Gaussian-function
• Weisstein, Eric W.Erf-function
• Weisstein, Eric W. "Hermite Polynomial".Hermite-polynomials
• OEISsequence A066325 (Coefficients of unitary Hermite polynomials He_n(x)) (Related to Gauss-matrix).

• Matrices
• Triangles of numbers

• Articles with short description
• Short description matches Wikidata
• Use American English from January 2019
• All Wikipedia articles written in American English