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Pascal matrices. (English) Zbl 1089.15025

Truncation in the Pascal’s triangle produces \(n\times n\) symmetric \(S_{n}\), lower triangular \(L_{n}\) and upper triangular \(U_{n}\) matrices. The relation among these three matrices is that for every \(n\), \(S_{n} =L_{n}U_{n}\). The paper offers four proofs of the equality \(S=LU\). Three of them are known and the fourth is new.
The first proof is based in the fact \(\sum L_{ik}U_{kj}=\sum_{k=0}^n \binom{i}{k}\binom{j}{k}= \binom{i+j}{i}=S_{ij}\). The second proof identifies \(S_{ij}\) as the number of paths between two points in a directed graph, \(L_{ik}\) as the number of paths between a point of the previous graph and the diagonal line and \(U_{kj}\) as the number of paths between the diagonal line and a point, and it is based in the fact that gluing the previous graphs is equivalent to multiplying \(L\) times \(U\). The third proof uses Gaussian elimination and Pascal’s recursion. The elimination matrix \(E\) with entries \(E_{ii}=1\) and \(E_{ii-1}=-1\)gives \(EL_{n}=\left( \begin{smallmatrix} 1 & 0\\ 0 & L_{n-1} \end{smallmatrix} \right) \). Using induction and that \((EL_{n})(U_{n}E^{T})=\left( \begin{smallmatrix} 1 & 0\\ 0 & L_{n-1} \end{smallmatrix} \right) \left( \begin{smallmatrix} 1 & 0\\ 0 & U_{n-1} \end{smallmatrix} \right)= \left( \begin{smallmatrix} 1 & 0\\ 0 & S_{n-1} \end{smallmatrix} \right) \), the proof is completed checking that \(ES_{n}E^{T}=\left( \begin{smallmatrix} 1 & 0\\ 0 & S_{n-1} \end{smallmatrix} \right) \).
The last proof, the favorite of the authors, verifies \(Sv=LUv\) for the family of vectors \(v=(1,x,x^{2},\ldots)\). For \(L\), \(U\) and \(S\) infinite matrices, they get \(Lv=\left( \begin{smallmatrix} 1\\ 1+x\\ (1+x)^{2}\\ \vdots \end{smallmatrix} \right) \), and if \(| x| <1\), \(Uv=\left( \begin{smallmatrix} 1/(1-x)\\ x/(1-x)^{2}\\ x^{2}/(1-x)^{3}\\ \vdots \end{smallmatrix} \right) \) and \(Sv=\left( \begin{smallmatrix} 1/(1-x)\\ 1/(1-x)^{2}\\ 1/(1-x)^{3}\\ \vdots \end{smallmatrix} \right) \), and thus \(Sv=LUv\) for all \(v=(1,x,x^{2},\ldots)\). The choice \(x=0\), \(v_{0}=(1,0,0,\ldots)\), gives agreement between the first columns of \(S\) and \(LU\), and derivative and higher derivatives of \(v\) at \(x=0\) give the other coordinate vectors, and so all the columns of \(S\) and \(LU\) are identical.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15B36 Matrices of integers
15A23 Factorization of matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C20 Directed graphs (digraphs), tournaments
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