In mathematics, particularly

matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \beg ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

, a Pascal matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

(possibly

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group) Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...

) containing the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s as its elements. It is thus an encoding of

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, althoug ...

in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an

upper-triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...

, or a

symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...

. For example, the 5 × 5 matrices are:

L_5 = \begin
1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
1 & 2 & 1 & 0 & 0 \\
1 & 3 & 3 & 1 & 0 \\
1 & 4 & 6 & 4 & 1
\end\,\,\,

U_5 = \begin
1 & 1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 & 4 \\
0 & 0 & 1 & 3 & 6 \\
0 & 0 & 0 & 1 & 4 \\
0 & 0 & 0 & 0 & 1
\end\,\,\,

S_5 = \begin
1 & 1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 & 5 \\
1 & 3 & 6 & 10 & 15 \\
1 & 4 & 10 & 20 & 35 \\
1 & 5 & 15 & 35 & 70
\end=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.

Definition

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

L_ =  = \frac, j \le i

U_ =  = \frac, i \le j

S_ =  =  = \frac

such that the indices ''i'', ''j'' start at 0, and ! denotes the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...

Properties

The matrices have the pleasing relationship ''S''_''n'' = ''L''_''n''''U''_''n''. From this it is easily seen that all three matrices have

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

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 :

\text(S_n) = \sum^n_ \frac = \sum^_ \frac

with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, … .

Construction

A Pascal matrix can actually be constructed by taking the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

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. :

\begin
& L_7=\exp
\left (
\left [
\begin
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 2 & . & . & . & . & . \\
. & . & 3 & . & . & . & . \\
. & . & . & 4 & . & . & . \\
. & . & . & . & 5 & . & . \\
. & . & . & . & . & 6 & .

\end
\right ]
\right )
=
\left [
\begin
1   & .   & .   & .   & .   & .   & .   \\
1   & 1   & .   & .   & .   & .   & .   \\
1   & 2   & 1   & .   & .   & .   & .   \\
1   & 3   & 3   & 1   & .   & .   & .   \\
1   & 4   & 6   & 4   & 1   & .   & .   \\
1   & 5   & 10  & 10  & 5   & 1   & .   \\
1   & 6   & 15  & 20  & 15  & 6   & 1  
\end
\right ]
;\quad
\\
\\
& U_7=\exp
\left (
\left [
\begin
. & 1 & . & . & . & . & . \\
. & . & 2 & . & . & . & . \\
. & . & . & 3 & . & . & . \\
. & . & . & . & 4 & . & . \\
. & . & . & . & . & 5 & . \\
. & . & . & . & . & . & 6 \\
. & . & . & . & . & . & .  
\end
\right ]
\right )
=
\left [
\begin
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
\right ]
;
\\
\\

\therefore & S_7
=\exp
\left (
\left [
\begin
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 2 & . & . & . & . & . \\
. & . & 3 & . & . & . & . \\
. & . & . & 4 & . & . & . \\
. & . & . & . & 5 & . & . \\
. & . & . & . & . & 6 & .

\end
\right ]
\right )
\exp
\left (
\left [
\begin
. & 1 & . & . & . & . & . \\
. & . & 2 & . & . & . & . \\
. & . & . & 3 & . & . & . \\
. & . & . & . & 4 & . & . \\
. & . & . & . & . & 5 & . \\
. & . & . & . & . & . & 6 \\
. & . & . & . & . & . & . 
\end
\right ]
\right )
=
\left [
\begin
1   & 1   & 1   & 1   & 1   & 1   & 1   \\
1   & 2   & 3   & 4   & 5   & 6   & 7   \\
1   & 3   & 6   & 10  & 15  & 21  & 28  \\
1   & 4   & 10  & 20  & 35  & 56  & 84  \\
1   & 5   & 15  & 35  & 70  & 126 & 210 \\
1   & 6   & 21  & 56  & 126 & 252 & 462 \\
1   & 7   & 28  & 84  & 210 & 462 & 924
\end
\right ].
\end

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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

; that is, when raised to a sufficiently great

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

power, they degenerate into the

zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed ...

. (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 In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, ma ...

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 :

\begin
& LAG_7=\exp
\left (
\left [
\begin
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 4 & . & . & . & . & . \\
. & . & 9 & . & . & . & . \\
. & . & . & 16 & . & . & . \\
. & . & . & . & 25 & . & . \\
. & . & . & . & . & 36 & .
\end
\right ]
\right )
=
\left [
\begin
    1 &      . &      . &      . &     . &    . &   .   \\
    1 &      1 &      . &      . &     . &    . &   .   \\
    2 &      4 &      1 &      . &     . &    . &   .   \\ 
    6 &     18 &      9 &      1 &     . &    . &   .   \\ 
   24 &     96 &     72 &     16 &     1 &    . &   .   \\ 
  120 &    600 &    600 &    200 &    25 &    1 &   .   \\ 
  720 &   4320 &   5400 &   2400 &   450 &   36 &   1 
\end
\right ]
;\quad
\end

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) :

\begin
& LAH_7 = \exp
\left (
\left [
\begin
. & . & . & . & . & . & . \\
2 & . & . & . & . & . & . \\
. & 6 & . & . & . & . & . \\
. & . &12 & . & . & . & . \\
. & . & . & 20 & . & . & . \\
. & . & . & . & 30 & . & . \\
. & . & . & . & . & 42 & .
\end
\right ]
\right )
=
\left [
\begin
 1 & . & . & . & . & . & . & . \\
 2 & 1 & . & . & . & . & . & . \\
 6 & 6 & 1 & . & . & . & . & . \\
 24 & 36 & 12 & 1 & . & . & . & . \\
 120 & 240 & 120 & 20 & 1 & . & . & . \\
 720 & 1800 & 1200 & 300 & 30 & 1 & . & . \\
 5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & . \\
 40320 & 141120 & 141120 & 58800 & 11760 & 1176 & 56 & 1
\end
\right ]
;\quad
\end

Using ''v''(''v'' − 1) instead provides a diagonal shifting to bottom-right. The third example below uses the square of the original ''PL''₇-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

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s and

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

s of the Gaussian

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...

: :

\begin
& GS_7 = \exp
\left (
\left [
\begin
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 3 & . & . & . & . & . \\
. & . & 6 & . & . & . & . \\
. & . & . & 10 & . & . & . \\
. & . & . & . & 15 & . & .
\end
\right ]
\right )
=
\left [
\begin
   1 &    . &    . &    . &    . &   . &   .   \\ 
   . &    1 &    . &    . &    . &   . &   .   \\ 
   1 &    . &    1 &    . &    . &   . &   .   \\ 
   . &    3 &    . &    1 &    . &   . &   .   \\ 
   3 &    . &    6 &    . &    1 &   . &   .   \\ 
   . &   15 &    . &   10 &    . &   1 &   .   \\ 
  15 &    . &   45 &    . &   15 &   . &   1 
\end
\right ]
;\quad
\end

If this matrix is inverse matrix, 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: :

\begin
& \exp
\left (
\left [
\begin
. & . & . & . & . & . & . & . & . & . & . & . \\
-5& . & . & . & . & . & . & . & . & . & . & . \\
. &-4 & . & . & . & . & . & . & . & . & . & . \\
. & . &-3 & . & . & . & . & . & . & . & . & . \\
. & . & . &-2 & . & . & . & . & . & . & . & . \\
. & . & . & . &-1 & . & . & . & . & . & . & . \\
. & . & . & . & . & 0 & . & . & . & . & . & . \\
. & . & . & . & . & . & 1 & . & . & . & . & . \\
. & . & . & . & . & . & . & 2 & . & . & . & . \\
. & . & . & . & . & . & . & . & 3 & . & . & . \\
. & . & . & . & . & . & . & . & . & 4 & . & . \\
. & . & . & . & . & . & . & . & . & . & 5 & .

\end
\right ]
\right )
=
\left [
\begin
1   & .   & .   & .   & .   & .   & .   & .   & .   & .   & .   & .   \\
-5  & 1   & .   & .   & .   & .   & .   & .   & .   & .   & .   & .   \\
10  & -4  & 1   & .   & .   & .   & .   & .   & .   & .   & .   & .   \\
-10 & 6   & -3  & 1   & .   & .   & .   & .   & .   & .   & .   & .   \\
5   & -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
\right ]
.
\end

References

* G. S. Call and D. J. Velleman, "Pascal's matrices", ''

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...

'', volume 100, (April 1993) pages 372–376 *

External links

* G. Helm
Pascalmatrix
in a project of compilation of facts abou

* G. Helm
Gauss-matrix
* Weisstein, Eric W

* Weisstein, Eric W

* Weisstein, Eric W. "Hermite Polynomial"

* Endl, Kurt "Über eine ausgezeichnete Eigenschaft der Koeffizientenmatrizen des Laguerreschen und des Hermiteschen Polynomsystems". In: PERIODICAL VOLUME 65 Mathematische Zeitschrif
Kurt Endl
* (Related to Gauss-matrix). {{Matrix classes Matrices Triangles of numbers