In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of vectors v 1 , … , v n {\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 G i j = ⟨ v i , v j ⟩ {\displaystyle G_{ij}=\left\langle v_{i},v_{j}\right\rangle } . If the vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} are the columns of matrix X {\displaystyle X} then the Gram matrix is X † X {\displaystyle X^{\dagger }X} in the general case that the vector coordinates are complex numbers, which simplifies to X ⊤ X {\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.
Examples For finite-dimensional real vectors in R n {\displaystyle \mathbb {R} ^{n}} with the usual Euclidean dot product, the Gram matrix is G = V ⊤ V {\displaystyle G=V^{\top }V} , where V {\displaystyle V} is a matrix whose columns are the vectors v k {\displaystyle v_{k}} and V ⊤ {\displaystyle V^{\top }} is its transpose whose rows are the vectors v k ⊤ {\displaystyle v_{k}^{\top }} . For complex vectors in C n {\displaystyle \mathbb {C} ^{n}} , G = V † V {\displaystyle G=V^{\dagger }V} , where V † {\displaystyle V^{\dagger }} is the conjugate transpose of V {\displaystyle V} . Given square-integrable functions { ℓ i ( ⋅ ) , i = 1 , … , n } {\displaystyle \{\ell _{i}(\cdot ),\,i=1,\dots ,n\}} on the interval [ t 0 , t f ] {\displaystyle \left[t_{0},t_{f}\right]} , the Gram matrix G = [ G i j ] {\displaystyle G=\left[G_{ij}\right]} is:
G i j = ∫ t 0 t f ℓ i ∗ ( τ ) ℓ j ( τ ) d τ . {\displaystyle G_{ij}=\int _{t_{0}}^{t_{f}}\ell _{i}^{*}(\tau )\ell _{j}(\tau )\,d\tau .}
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