Maths - Tensor - Gram Matrix

The Gram Matrix is also known as the Gramian matrix.

The Gram matrix of a set of vectors is the symmetric matrix of inner products.

The entries are given by G_ij = v_i^T v_j = (v_i | v_j).

viT vj =

a0 a1 b0 b1 = a0*b0 a0*b1 a1*b0 a1*b1

 

 

A set of vectors is linearly independent if and only if: det(G_ij) ≠ 0

 

Gram Matrix as the covariance matrix

if e_i represents the basis vectors then:

e^*i = G_ij e_i

represents the reciprocal or dual basis.

If the basis is orthogonal then G_ij = δ_ij = identity matrix and so the the basis and its dual are effectively equal.

 

EuclideanSpace

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