Gramian matrix

In linear algebra, the Gram matrix (Gramian matrix or 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 ${\displaystyle G_{ij}=\langle v_{i},v_{j}\rangle }$.

An important application is to compute linear independence: a set of vectors is 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.

For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply ${\displaystyle G=V^{\mathrm {T} }V}$ (or ${\displaystyle G=V^{H}V}$ for complex vectors using the conjugate transpose), where V is a matrix whose columns are the vectors ${\displaystyle v_{k}}$.

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