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 Gij = viT vj = (vi | vj).

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(Gij) ≠ 0

 

Gram Matrix as the covariance matrix

if ei represents the basis vectors then:

e*i = Gij ei

represents the reciprocal or dual basis.

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

 


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