@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-JR0BZJDR-C skos:prefLabel "square matrix"@en, "matrice carrée"@fr ; a skos:Concept ; skos:narrower psr:-B268B0H3-3 . psr: a skos:ConceptScheme . psr:-B268B0H3-3 skos:definition """En algèbre linéaire et multilinéaire, une matrice symétrique est une matrice carrée qui est égale à sa propre transposée, c'est-à-dire telle que a_i,j = a_j,i pour tous i et j compris entre 1 et n, où les a_i,j sont les coefficients de la matrice et n est son ordre.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Matrice_sym%C3%A9trique)"""@fr, """In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,

{\\\\displaystyle A{\\ ext{ is symmetric}}\\\\iff A=A^{\\ extsf {T}}.}

Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if

{\\\\displaystyle a_{ij}}

denotes the entry in the

{\\\\displaystyle i}

th row and

{\\\\displaystyle j}

th column then

{\\\\displaystyle A{\\ ext{ is symmetric}}\\\\iff {\\ ext{ for every }}i,j,\\\\quad a_{ji}=a_{ij}}

for all indices

{\\\\displaystyle i}

and

{\\\\displaystyle j.}

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Symmetric_matrix)"""@en ; skos:exactMatch , ; skos:prefLabel "matrice symétrique"@fr, "symmetric matrix"@en ; skos:broader psr:-JR0BZJDR-C ; a skos:Concept ; skos:inScheme psr: ; dc:modified "2024-10-18"^^xsd:date .