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In linear algebra, the Gram matrix (Gramian matrix or Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by .[1]

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.

Contents

ExamplesEdit

For finite-dimensional real vectors in   with the usual Euclidean dot product, the Gram matrix is simply  , where   is a matrix whose columns are the vectors  . For complex vectors in  ,  , where   is the conjugate transpose of  .

Given square-integrable functions   on the interval  , the Gram matrix   is:

 

For any bilinear form   on a finite-dimensional vector space over any field we can define a Gram matrix   attached to a set of vectors   by  . The matrix will be symmetric if the bilinear form   is symmetric.

ApplicationsEdit

  • In Riemannian geometry, given an embedded  -dimensional Riemannian manifold   and a coordinate chart   for  , the volume form   on   induced by the embedding may be computed using the Gramian of the coordinate tangent vectors:
 

This generalizes the classical surface integral of a parametrized surface   for  :

 

PropertiesEdit

Positive-semidefinitenessEdit

The Gramian matrix is positive-semidefinite, and every positive symmetric semidefinite matrix is the Gramian matrix for some set of vectors. Further, in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix. These facts follow from taking the spectral decomposition of any positive-semidefinite matrix  , so that   and so   is the Gramian matrix of the rows of  . The Gramian matrix of any orthonormal basis is the identity matrix. The infinite-dimensional analog of this statement is Mercer's theorem.

Derivation of positive-semidefinitenessEdit

The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:

 

The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of the inner-product, and the last from the positive definiteness of the inner product.

Note that this also shows that the Gramian matrix is positive definite if and only if the vectors   are linearly independent.

Gram determinantEdit

The Gram determinant or Gramian is the determinant of the Gram matrix:

 

If   are vectors in  , then it is the square of the n-dimensional volume of the parallelotope formed by the vectors. In particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular.

The Gram determinant can also be expressed in terms of the exterior product of vectors by

 

See alsoEdit

ReferencesEdit

  1. ^ Horn & Johnson 2013, p. 441
    Theorem 7.2.10 Let   be vectors in an inner product space   with inner product   and let  . Then
    (a)   is Hermitian and positive-semidefinite
    (b)   is positive-definite if and only if the vectors   are linearly-independent.
    (c)  
  2. ^ Lanckriet, G. R. G.; Cristianini, N.; Bartlett, P.; Ghaoui, L. E.; Jordan, M. I. (2004). "Learning the kernel matrix with semidefinite programming". Journal of Machine Learning Research. 5: 27–72 [p. 29].

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