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{{Short description|Infinite matrices with Pascal's triangle as elements}}
In [[Mathematics|mathematics]], particularly [[Matrix (mathematics)|matrix theory]] and [[Combinatorics|combinatorics]], a '''Pascal matrix''' is a [[Matrix (mathematics)|matrix]] (possibly [[Matrix (mathematics)#Infinite matrices|infinite]]) containing the [[Binomial coefficient|binomial coefficient]]s 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|symmetric matrix]]. For example, the 5&thinsp;×&thinsp;5 matrices are:

<math display="block">L_5 = \begin{pmatrix}
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{pmatrix}\,\,\,</math><!-- PLEASE do not remove the \,\,\, - it is there to force a higher quality anti-aliasing mode in the wikitex renderer --><math display="block">U_5 = \begin{pmatrix}
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{pmatrix}\,\,\,</math><!-- PLEASE do not remove the \,\,\, - it is there to force a higher quality anti-aliasing mode in the wikitex renderer --><math display="block">S_5 = \begin{pmatrix}
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{pmatrix}=L_5 \times U_5</math>
There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity.<ref>{{Cite journal|date=2010-07-01|title=A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations|journal=European Journal of Combinatorics|language=en|volume=31|issue=5|pages=1205–1216|doi=10.1016/j.ejc.2009.10.009|issn=0195-6698|doi-access=|last1=Birregah |first1=Babiga |last2=Doh |first2=Prosper K. |last3=Adjallah |first3=Kondo H. }}</ref>

== Definition ==

The non-zero elements of a Pascal matrix are given by the [[Binomial coefficient|binomial coefficient]]s:

<math display="block">L_{ij} = {i \choose j} = \frac{i!}{j!(i-j)!}, j \le i</math>
<math display="block">U_{ij} = {j \choose i} = \frac{j!}{i!(j-i)!}, i \le j</math>
<math display="block">S_{ij} = {i+j \choose i} = {i+j \choose j} = \frac{(i+j)!}{i!j!}</math>

such that the indices ''i'', ''j'' start at 0, and ! denotes the [[Factorial|factorial]].

== Properties ==
The matrices have the pleasing relationship ''S''<sub>''n''</sub> = ''L''<sub>''n''</sub>''U''<sub>''n''</sub>. From this it is easily seen that all three matrices have [[Determinant|determinant]] 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for both ''L<sub>n</sub>'' and ''U''<sub>''n''</sub>. In other words, matrices ''S''<sub>''n''</sub>, ''L''<sub>''n''</sub>, and ''U''<sub>''n''</sub> are [[Unimodular matrix|unimodular]], with ''L''<sub>''n''</sub> and ''U''<sub>''n''</sub> having trace ''n''.

The trace of ''S<sub>n</sub>'' is given by
:<math>\text{tr}(S_n) = \sum^n_{i=1} \frac{ [ 2(i-1) ] !}{[(i-1)!]^2} = \sum^{n-1}_{k=0} \frac{ (2k) !}{(k!)^2}</math>
with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... {{OEIS|A006134}}.

==Construction==
A Pascal matrix can actually be constructed by taking the [[Matrix exponential|matrix exponential]] of a special [[Diagonal#Matrices|subdiagonal]] or [[Diagonal#Matrices|superdiagonal]] matrix. The example below constructs a 7&thinsp;×&thinsp;7 Pascal matrix, but the method works for any desired ''n''&thinsp;×&thinsp;''n'' Pascal matrices. The dots in the following matrices represent zero elements.

:<math>
\begin{array}{lll}
& L_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 2 & . & . & . & . & . \\
. & . & 3 & . & . & . & . \\
. & . & . & 4 & . & . & . \\
. & . & . & . & 5 & . & . \\
. & . & . & . & . & 6 & .

\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
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{smallmatrix}
\right ]
;\quad
\\
\\
& U_7=\exp
\left (
\left [
\begin{smallmatrix}
{\color{white}1}. & 1 & . & . & . & . & . \\
{\color{white}1}. & . & 2 & . & . & . & . \\
{\color{white}1}. & . & . & 3 & . & . & . \\
{\color{white}1}. & . & . & . & 4 & . & . \\
{\color{white}1}. & . & . & . & . & 5 & . \\
{\color{white}1}. & . & . & . & . & . & 6 \\
{\color{white}1}. & . & . & . & . & . & .  
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
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{smallmatrix}
\right ]
;
\\
\\

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

\end{smallmatrix}
\right ]
\right )
\exp
\left (
\left [
\begin{smallmatrix}
{\color{white}i}. & 1 & . & . & . & . & . \\
{\color{white}i}. & . & 2 & . & . & . & . \\
{\color{white}i}. & . & . & 3 & . & . & . \\
{\color{white}i}. & . & . & . & 4 & . & . \\
{\color{white}i}. & . & . & . & . & 5 & . \\
{\color{white}i}. & . & . & . & . & . & 6 \\
{\color{white}i}. & . & . & . & . & . & . 
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
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{smallmatrix}
\right ].
\end{array}
</math>

It is important to note that one cannot simply assume exp(''A'')&thinsp;exp(''B'') = exp(''A''&nbsp;+&nbsp;''B''), for ''n''&thinsp;×&thinsp;''n'' matrices ''A'' and ''B''; this equality is only true when ''AB'' = ''BA'' (i.e. when the matrices ''A'' and ''B'' [[Commuting matrices|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|nilpotent]]; that is, when raised to a sufficiently great [[Integer|integer]] power, they degenerate into the [[Zero matrix|zero matrix]]. (See [[Shift matrix|shift matrix]] for further details.) As the ''n''&thinsp;×&thinsp;''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''&nbsp;+&thinsp;1 terms of the infinite series to obtain an exact result.

==Variants==
Interesting variants can be obtained by obvious modification of the matrix-logarithm PL<sub>7</sub> 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&thinsp;×&thinsp;7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials
:<math>
\begin{array}{lll}
& LAG_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 4 & . & . & . & . & . \\
. & . & 9 & . & . & . & . \\
. & . & . & 16 & . & . & . \\
. & . & . & . & 25 & . & . \\
. & . & . & . & . & 36 & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
    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{smallmatrix}
\right ]
;\quad
\end{array}
</math>

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''&nbsp;+&thinsp;1) of the values of the log-matrix and constructs a 7&thinsp;×&thinsp;7 "Lah"- matrix  (or matrix of coefficients of [[Lah number]]s)

:<math>\begin{array}{lll}
& LAH_7 = \exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
2 & . & . & . & . & . & . \\
. & 6 & . & . & . & . & . \\
. & . &12 & . & . & . & . \\
. & . & . & 20 & . & . & . \\
. & . & . & . & 30 & . & . \\
. & . & . & . & . & 42 & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
 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{smallmatrix}
\right ]
;\quad
\end{array}
</math>
Using ''v''(''v''&nbsp;&minus;&nbsp;1) instead provides a diagonal shifting to bottom-right.

The third example below uses the square of the original ''PL''<sub>7</sub>-matrix, divided by 2, in other words: the first-order binomials (binomial(''k'',&nbsp;2)) in the second subdiagonal and constructs a matrix, which occurs in context of the [[Derivative|derivative]]s and [[Integral|integral]]s of the Gaussian [[Error function|error function]]:

:<math>
\begin{array}{lll}
& GS_7 = \exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 3 & . & . & . & . & . \\
. & . & 6 & . & . & . & . \\
. & . & . & 10 & . & . & . \\
. & . & . & . & 15 & . & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
   1 &    . &    . &    . &    . &   . &   .   \\ 
   . &    1 &    . &    . &    . &   . &   .   \\ 
   1 &    . &    1 &    . &    . &   . &   .   \\ 
   . &    3 &    . &    1 &    . &   . &   .   \\ 
   3 &    . &    6 &    . &    1 &   . &   .   \\ 
   . &   15 &    . &   10 &    . &   1 &   .   \\ 
  15 &    . &   45 &    . &   15 &   . &   1 
\end{smallmatrix}
\right ]
;\quad
\end{array}
</math>
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 [[Pascal's triangle#Extensions|negative value]]s:  
:<math>
\begin{array}{lll}
& \exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . & . & . & . & . & . \\
-5& . & . & . & . & . & . & . & . & . & . & . \\
. &-4 & . & . & . & . & . & . & . & . & . & . \\
. & . &-3 & . & . & . & . & . & . & . & . & . \\
. & . & . &-2 & . & . & . & . & . & . & . & . \\
. & . & . & . &-1 & . & . & . & . & . & . & . \\
. & . & . & . & . & 0 & . & . & . & . & . & . \\
. & . & . & . & . & . & 1 & . & . & . & . & . \\
. & . & . & . & . & . & . & 2 & . & . & . & . \\
. & . & . & . & . & . & . & . & 3 & . & . & . \\
. & . & . & . & . & . & . & . & . & 4 & . & . \\
. & . & . & . & . & . & . & . & . & . & 5 & .

\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
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{smallmatrix}
\right ]
.
\end{array}
</math>

==See also==
*[[Pascal's triangle]]
*[[LU decomposition]]
* [[Riordan array|Riordan Array]]

==References==
{{reflist}}
* G. S. Call and D. J. Velleman, "Pascal's matrices", ''[[American Mathematical Monthly]]'', volume 100, (April 1993) pages 372&ndash;376
* {{Citation | first1 = Alan | last1 = Edelman | first2 = Gilbert | last2 = Strang |  title = Pascal Matrices | journal = [[American Mathematical Monthly]] | volume = 111 | number = 3 | date = March 2004 | pages = 361&ndash;385 | url = http://mathdl.maa.org/images/upload_library/22/Ford/Edelman189-197.pdf | doi = 10.2307/4145127 | jstor = 4145127 | url-status = dead | archiveurl = https://web.archive.org/web/20100704210817/http://mathdl.maa.org/images/upload_library/22/Ford/Edelman189-197.pdf | archivedate = 2010-07-04 }}

==External links==
* G. Helms [https://go.helms-net.de/math/binomial_new/01_1_binomialmatrix.pdf Pascalmatrix] in a project of compilation of facts about [https://go.helms-net.de/math/binomial/index.htm Numbertheoretical matrices]
* G. Helms [https://go.helms-net.de/math/binomial/04_1_gaussmatrix.pdf Gauss-matrix]
* Weisstein, Eric W. [https://mathworld.wolfram.com/GaussianFunction.html Gaussian-function]  
* Weisstein, Eric W. [https://mathworld.wolfram.com/Erf.html Erf-function] 
* Weisstein, Eric W. "Hermite Polynomial". [https://mathworld.wolfram.com/HermitePolynomial.html Hermite-polynomials] 
* Endl, Kurt "Über eine ausgezeichnete Eigenschaft der Koeffizientenmatrizen des Laguerreschen und des Hermiteschen Polynomsystems". In: PERIODICAL VOLUME 65 Mathematische Zeitschrift [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0065 Kurt Endl] 
* {{OEIS el|1=A066325|2=Coefficients of unitary Hermite polynomials He<sub>''n''</sub>(''x'')|formalname=Coefficients of unitary Hermite polynomials He_n(x)}} (Related to Gauss-matrix).

{{Matrix classes}}

[[Category:Matrices]]
[[Category:Triangles of numbers]]

{{Sourceattribution|Pascal matrix}}