Gramian matrix

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In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by G_{ij}=\langle v_j, v_i \rangle. For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply G = V^\mathrm{T} V (or G = V^\dagger V for complex vectors using the conjugate transpose), where V is a matrix whose columns are the vectors v_k.

An important application is to compute linear independence: a set of vectors is 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.

Examples[edit]

Most commonly, the vectors are elements of an Euclidean space, or are functions in an L2 space, such as continuous functions on a compact interval [ab] (which are a subspace of L 2([ab])).

Given real-valued functions \{\ell_i(\cdot),\,i=1,\dots,n\} on the interval [t_0,t_f], the Gram matrix G=[G_{ij}], is given by the standard inner product on functions:

G_{ij}=\int_{t_0}^{t_f} \ell_i(\tau)\bar{\ell_j}(\tau)\, d\tau.

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 v_1,\dots, v_n by G_{ij} = B(v_i,v_j) \, . The matrix will be symmetric if the bilinear form B is symmetric.

Applications[edit]

Properties[edit]

Positive semidefinite[edit]

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 P = U D U^* = (U\sqrt{D}) (U \sqrt{D})^* and so P is the Gramian matrix of the columns of U\sqrt{D}. 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[edit]

Under change of basis represented by an invertible matrix P, the Gram matrix will change by a matrix congruence to PTGP.

Gram determinant[edit]

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

G(x_1,\dots, x_n)=\begin{vmatrix} \langle x_1,x_1\rangle & \langle x_1,x_2\rangle &\dots & \langle x_1,x_n\rangle\\
 \langle x_2,x_1\rangle & \langle x_2,x_2\rangle &\dots & \langle x_2,x_n\rangle\\
\vdots&\vdots&\ddots&\vdots\\
 \langle x_n,x_1\rangle & \langle x_n,x_2\rangle &\dots & \langle x_n,x_n\rangle\end{vmatrix}.

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

G(x_1,\dots,x_n) = \| x_1\wedge\cdots\wedge x_n\|^2.

See also[edit]

References[edit]

  1. ^ 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.

External links[edit]