Editing Gram matrix (section)

{{short description|Matrix of inner products of a set of vectors}}
In [[linear algebra]], the '''Gram matrix''' (or '''Gramian matrix''', '''Gramian''') of a set of vectors <math>v_1,\dots, v_n</math> in an [[inner product space]] is the [[Hermitian matrix]] of [[inner product]]s, whose entries are given by the [[inner product]] <math>G_{ij} = \left\langle v_i, v_j \right\rangle</math>.<ref name="HJ-7.2.10">{{harvnb|Horn|Johnson|2013|p=441}}, p.441, Theorem 7.2.10</ref> If the vectors <math>v_1,\dots, v_n</math> are the columns of matrix <math>X</math> then the Gram matrix is <math>X^\dagger X</math> in the general case that the vector coordinates are complex numbers, which simplifies to <math>X^\top X</math> for the case that the vector coordinates are real numbers.

An important application is to compute [[linear independence]]: a set of vectors are linearly independent if and only if the [[#Gram determinant|Gram determinant]] (the [[determinant]] of the Gram matrix) is non-zero.

It is named after [[Jørgen Pedersen Gram]].