Gram matrix

In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors ${\displaystyle v_{1},\dots ,v_{n}}$ in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product ${\displaystyle G_{ij}=\left\langle v_{i},v_{j}\right\rangle }$.^[1] If the vectors ${\displaystyle v_{1},\dots ,v_{n}}$ are the columns of matrix ${\displaystyle X}$ then the Gram matrix is ${\displaystyle X^{\dagger }X}$ in the general case that the vector coordinates are complex numbers, which simplifies to ${\displaystyle X^{\top }X}$ 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 (the determinant of the Gram matrix) is non-zero.

It is named after Jørgen Pedersen Gram.