Gramian matrix

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

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.

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  :

     
  • If the vectors are centered random variables, the Gramian is approximately proportional to the covariance matrix, with the scaling determined by the number of elements in the vector.
  • In quantum chemistry, the Gram matrix of a set of basis vectors is the overlap matrix.
  • In control theory (or more generally systems theory), the controllability Gramian and observability Gramian determine properties of a linear system.
  • Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp. 79–94).
  • In the finite element method, the Gram matrix arises from approximating a function from a finite dimensional space; the Gram matrix entries are then the inner products of the basis functions of the finite dimensional subspace.
  • In machine learning, kernel functions are often represented as Gram matrices.[2]
  • Since the Gram matrix over the reals is a symmetric matrix, it is diagonalizable and its eigenvalues are non-negative. The diagonalization of the Gram matrix is the singular value decomposition.

PropertiesEdit

Positive-semidefinitenessEdit

The Gram matrix is symmetric in the case the real product is real-valued; it is Hermitian in the general, complex case by definition of an inner product.

The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. 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 (that is,   for all  ).[1]

Finding a vector realizationEdit

Given any positive semidefinite matrix  , one can decompose it as:

 ,

where   is the conjugate transpose of   (or   in the real case).

Here   is a   matrix, where   is the rank of  . Various ways to obtain such a decomposition include computing the Cholesky decomposition or taking the non-negative square root of  .

The columns   of   can be seen as n vectors in   (or k-dimensional Euclidean space  , in the real case). Then

 

where the dot product   is the usual inner product on  .

Thus a Hermitian matrix   is positive semidefinite if and only if it is the Gram matrix of some vectors  . Such vectors are called a vector realization of  . The infinite-dimensional analog of this statement is Mercer's theorem.

Uniqueness of vector realizationsEdit

If   is the Gram matrix of vectors   in  , then applying any rotation or reflection of   (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any   orthogonal matrix  , the Gram matrix of   is also  .

This is the only way in which two real vector realizations of   can differ: the vectors   are unique up to orthogonal transformations. In other words, the dot products   and   are equal if and only if some rigid transformation of   transforms the vectors   to   and 0 to 0.

The same holds in the complex case, with unitary transformations in place of orthogonal ones. That is, if the Gram matrix of vectors   is equal to the Gram matrix of vectors   in  , then there is a unitary   matrix   (meaning  ) such that   for  .[3]

Other propertiesEdit

  • The Gram matrix of any orthonormal basis is the identity matrix.
  • The rank of the Gram matrix of vectors in   or   equals the dimension of the space spanned by these vectors.[1]

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. When m=n, This reduces to the standard theorem that determinant of n n-dimensional vectors is the n dimensional volume.

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

 

See alsoEdit

ReferencesEdit

  1. ^ a b c Horn & Johnson 2013, p. 441, p.441, Theorem 7.2.10
  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].
  3. ^ Horn & Johnson (2013), p. 452, Theorem 7.3.11

External linksEdit