In linear algebra, the Gramian matrix (or Gram 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 . For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply (or for complex vectors using the conjugate transpose), where V is a matrix whose columns are the vectors .
It is named after Jørgen Pedersen Gram.
Given a real matrix A, the matrix ATA is a Gram matrix (of the columns of A), while the matrix AAT is the Gram matrix of the rows of A.
For a general bilinear form B on a finite-dimensional vector space over any field we can define a Gram matrix G attached to a set of vectors by . The matrix will be symmetric if the bilinear form B is symmetric.
- If the vectors are centered random variables, the Gramian is 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, observability Gramian and cross 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.
The Gramian matrix is positive semidefinite, and every positive 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 P, so that and so P is the Gramian matrix of the columns of . The Gramian matrix of any orthonormal basis is the identity matrix. The infinite-dimensional analog of this statement is Mercer's theorem.
Change of basis
Under change of basis represented by an invertible matrix P, the Gram matrix will change by a matrix congruence to PTGP.
The Gram determinant or Gramian is the determinant of the Gram matrix:
Geometrically, the Gram determinant is the square of the volume of the parallelotope formed by the vectors. In particular, the vectors are linearly independent if and only if the 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
- Lanckriet, G. R. G., N. Cristianini, P. Bartlett, L. E. Ghaoui, and M. I. Jordan. “Learning the kernel matrix with semidefinite programming.” The Journal of Machine Learning Research 5 (2004): 29.
- Barth, Nils (1999). "The Gramian and K-Volume in N-Space: Some Classical Results in Linear Algebra". Journal of Young Investigators 2.