Pascal matrix

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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:

L5=(1000011000121001331014641)U5=(1111101234001360001400001)S5=(111111234513610151410203515153570)=L5×U5 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:

Lij=(ij)=i!j!(i−j)!,j≤i Uij=(ji)=j!i!(j−i)!,i≤j Sij=(i+ji)=(i+jj)=(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

tr(Sn)=∑i=1n[2(i−1)]![(i−1)!]2=∑k=0n−1(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.

L7=exp⁡([.......1.......2.......3.......4.......5.......6.])=[1......11.....121....1331...14641..15101051.1615201561];U7=exp⁡([1.1.....1..2....1...3...1....4..1.....5.1......61.......])=[1111111.123456..1361015...141020....1515.....16......1];∴S7=exp⁡([.......1.......2.......3.......4.......5.......6.])exp⁡([i.1.....i..2....i...3...i....4..i.....5.i......6i.......])=[111111112345671361015212814102035568415153570126210162156126252462172884210462924].

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

LAG7=exp⁡([.......1.......4.......9.......16.......25.......36.])=[1......11.....241....61891...249672161..120600600200251.720432054002400450361];

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)

LAH7=exp⁡([.......2.......6.......12.......20.......30.......42.])=[1.......21......661.....2436121....120240120201...72018001200300301..504015120126004200630421.4032014112014112058800117601176561];

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:

GS7=exp⁡([..............1.......3.......6.......10.......15..])=[1.......1.....1.1.....3.1...3.6.1...15.10.1.15.45.15.1];

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:

exp⁡([............−5............−4............−3............−2............−1............0............1............2............3............4............5.])=[1...........−51..........10−41.........−106−31........5−43−21.......−11−11−11...........01...........11..........121.........1331........14641.......15101051].

See also

References

  1. ↑ 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" (in en). European Journal of Combinatorics 31 (5): 1205–1216. doi:10.1016/j.ejc.2009.10.009. ISSN 0195-6698. 



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