1. A square matrix whose entries are the inner products of vectors, used in linear algebra and functional analysis to determine linear independence and compute volumes
The Gram matrix of a set of vectors v1, v2, ..., vn has entries G_ij = ⟨vi, vj⟩, where ⟨·,·⟩ denotes the inner product.
A matriz de Gram de um conjunto de vetores v1, v2, ..., vn tem entradas G_ij = ⟨vi, vj⟩, onde ⟨·,·⟩ denota o produto interno.
2. A symmetric positive semidefinite matrix constructed from the Gram determinant, useful for analyzing geometric properties of vector sets
The Gram matrix is always symmetric and its determinant equals the square of the volume of the parallelepiped formed by the vectors.
A matriz de Gram é sempre simétrica e seu determinante é igual ao quadrado do volume do paralelepípedo formado pelos vetores.
The Gram matrix is a fundamental concept in linear algebra and numerical analysis, named after Danish mathematician Jørgen Pedersen Gram. It is universally used in both Brazilian and European Portuguese mathematical literature without significant regional variation. The term is typically not translated in academic papers and research, maintaining the English designation 'Gram matrix' even in Portuguese-language publications.