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
tracen.
The trace of Sn is given by
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.
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 Bcommute). 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
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)
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 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:
^Birregah, Babiga; Doh, Prosper K.; Adjallah, Kondo H. (2010-07-01). "A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations". European Journal of Combinatorics. 31 (5): 1205–1216.
doi:
10.1016/j.ejc.2009.10.009.
ISSN0195-6698.
G. S. Call and D. J. Velleman, "Pascal's matrices", American Mathematical Monthly, volume 100, (April 1993) pages 372–376
Endl, Kurt "Über eine ausgezeichnete Eigenschaft der Koeffizientenmatrizen des Laguerreschen und des Hermiteschen Polynomsystems". In: PERIODICAL VOLUME 65 Mathematische Zeitschrift
Kurt Endl