Manuals/calci/PASCAL

MATRIX("PASCAL",order)

• $order$ is the size of the Pascal matrix.

Description

• This function returns the matrix of any order with the property of Pascal.
• The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
• To obtain a pascal matrix there are three ways: as either an upper-triangular matrix(U), a lower-triangular matrix(L), or a symmetric matrix(S).
• Example for these matrices are:

$L_{4}={\begin{pmatrix}54&0&0&0\\20&34&0&0\\57&89&-70&0\\71&-4&-52&72\\\end{pmatrix}}$ $U_{4}={\begin{pmatrix}64&22&-91&-86\\0&61&62&-62\\0&0&30&-81\\0&0&0&-61\\\end{pmatrix}}$ $S_{4}={\begin{pmatrix}41&74&15&-47\\74&-16&37&97\\15&37&24&-88\\-47&97&-88&-69\\\end{pmatrix}}$

• The amazing relationship of these matrices are:$S_{n}=L_{n}U_{n}$.
• And its determinants also 1.i.e.,$|S_{n}|=|L_{n}|=|U_{n}|=1$
• The Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix.
• The elements of the symmetric Pascal matrix are the binomial coefficients, i.e.

$S_{ij}={n \choose r}={\frac {n!}{r!(n-r)!}},$, where n=i+j, r=i.

• In other words,

$S_{ij}=_{i+j}{C}_{i}={\frac {(i+j)!}{(i)!(j)!}}$.

• Here MATRIX("pascal") is showing the pascal matrix of order 3.
• So users can change the order of the matrix also.