@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-KSXWNH06-H skos:prefLabel "matrix"@en, "matrice"@fr ; a skos:Concept ; skos:narrower psr:-FK57VT9K-Z . psr: a skos:ConceptScheme . psr:-FK57VT9K-Z skos:definition """In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an

{\\\\displaystyle (m+1)\\ imes (n+1)}

matrix

{\\\\displaystyle V=V(x_{0},x_{1},\\\\cdots ,x_{m})={\\egin{bmatrix}1&x_{0}&x_{0}^{2}&\\\\dots &x_{0}^{n}\\\\\\\\1&x_{1}&x_{1}^{2}&\\\\dots &x_{1}^{n}\\\\\\\\1&x_{2}&x_{2}^{2}&\\\\dots &x_{2}^{n}\\\\\\\\\\\\vdots &\\\\vdots &\\\\vdots &\\\\ddots &\\\\vdots \\\\\\\\1&x_{m}&x_{m}^{2}&\\\\dots &x_{m}^{n}\\\\end{bmatrix}}}

with entries

{\\\\displaystyle V_{i,j}=x_{i}^{j}}

, the j^th power of the number

{\\\\displaystyle x_{i}}

, for all zero-based indices

{\\\\displaystyle i}

and

{\\\\displaystyle j}

. Most authors define the Vandermonde matrix as the transpose of the above matrix
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Vandermonde_matrix)"""@en, """En algèbre linéaire, une matrice de Vandermonde est une matrice avec une progression géométrique dans chaque ligne. Elle tient son nom du mathématicien français Alexandre-Théophile Vandermonde. De façon matricielle, elle se présente ainsi :

{\\\\displaystyle V={\\egin{pmatrix}1&\\\\alpha _{1}&{\\\\alpha _{1}}^{2}&\\\\dots &{\\\\alpha _{1}}^{n-1}\\\\\\\\1&\\\\alpha _{2}&{\\\\alpha _{2}}^{2}&\\\\dots &{\\\\alpha _{2}}^{n-1}\\\\\\\\1&\\\\alpha _{3}&{\\\\alpha _{3}}^{2}&\\\\dots &{\\\\alpha _{3}}^{n-1}\\\\\\\\\\\\vdots &\\\\vdots &\\\\vdots &&\\\\vdots \\\\\\\\1&\\\\alpha _{m}&{\\\\alpha _{m}}^{2}&\\\\dots &{\\\\alpha _{m}}^{n-1}\\\\\\\\\\\\end{pmatrix}}}

Autrement dit, pour tous i et j, le coefficient en ligne i et colonne j est

{\\\\displaystyle V_{i,j}={\\\\alpha _{i}}^{j-1}.}