Pascal matrix

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, a lower-triangular matrix, or a symmetric matrix. The 5×5 truncations of these are shown below.

Upper triangular:  lower triangular:  symmetric:

These 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 elements of the symmetric Pascal matrix are the binomial coefficients, i.e.

In other words,

Thus the trace of S_n is given by

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

Contents 1 Construction 2 Variants 3 See also 4 References 5 External links

Construction

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

It is important to note that one cannot simply assume exp(A)exp(B) = exp(A + B), for A and B n×n matrices. Such an identity only holds when AB = BA (i.e. 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 in the construction is that both are nilpotent; that is, when raised to a sufficiently high 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-by-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-by-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 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:

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 the Gauss' error-function . (Literature about generalizations to higher powers is not found yet.)

See also

• Pascal's triangle
• LU decomposition

References

• G. S. Call and D. J. Velleman, "Pascal's matrices", American Mathematical Monthly, volume 100, (April 1993) pages 372–376
• Edelman, Alan; Strang, Gilbert (March 2004), "Pascal Matrices", American Mathematical Monthly 111 (3): 361–385, http://mathdl.maa.org/images/upload_library/22/Ford/Edelman189-197.pdf 

External links

• G. Helms Pascalmatrix in a project of compilation of facts about binomial&related matrices
• G. Helms Gauss-matrix

• Weisstein, Eric W. Gaussian-function
• Weisstein, Eric W. Erf-function
• Weisstein, Eric W. "Hermite Polynomial." Hermite-polynomials
• Endl, Kurt "Über eine ausgezeichnete Eigenschaft der Koeffizientenmatrizen des Laguerreschen und des Hermiteschen Polynomsystems". In: PERIODICAL VOLUME 65 Mathematische Zeitschrift Kurt Endl
• "Coefficients of unitary Hermite polynomials He_n(x)" in the "Online Encyclopedia of Integer Sequences" (OEIS) A066325 (Related to Gauss-matrix).