Gram matrix

For a vector spaceMathworldPlanetmath V of dimensionPlanetmathPlanetmath n over a field k, endowed with an inner product <.,.>:V×Vk, and for any given sequenceMathworldPlanetmath of elementsMathworldMathworld x1,,xlV, consider the following inclusion mapMathworldPlanetmath ι associated to the (xi)i=1,,l:


The Gram bilinear form of the (xi)i=1,,l is the function


The Gram matrix of the (xi)i=1,,l is the matrix associated to the Gram bilinear form in the canonical basis of kl. The Gram form (resp. matrix) is a symmetric bilinear formMathworldPlanetmath (resp. matrix).

Gram forms/matrices are usually considered with k= and <.,.> the usual scalar productsMathworldPlanetmath on vector spaces over . In that context they have a strong geometrical meaning:

  • The determinantMathworldPlanetmath of the Gram form/matrix is 0 iff ι is an injection.

  • If ι is injective, the Gram matrix (resp. form) is a positivePlanetmathPlanetmath symmetric matrixMathworldPlanetmath (resp. bilinear formPlanetmathPlanetmath).

  • det(G)-2=Vol(ι-1(B0,1)) where G is the gram form/matrix, Vol denotes the volume of a subset of n, and B0,1 is the unit ball of n centered at 0.

  • Let Δ:λlG(λ,λ), where G is the Gram bilinear form, then ι-1(B0,1)=Δ-1([0,1]).

  • If ι is injective and s:Im(ι)nl is an isometry, then det(sι)2=det(G).

  • Let f be an endomorphism of n and M its matrix. Let H,U be the polar decomposition of M, H is a symmetricPlanetmathPlanetmathPlanetmath positive matrix and U an orthogonal matrixMathworldPlanetmath. Let the xi be the columns of M (l=n) and let G be the Gram matrix the xi. Then H2=G. (N.B.: this is one way to prove the existence of the polar decomposition, take the square root of the Gram matrix, multiply M by its inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and it easily follows that what is obtained is an orthogonal matrix).

They are utilized in statistics in Principal components analysis. One wants to determine the general trend in terms of few characteristics (l of them) in a large sample (n individuals). Each xi(n) represents the results of the n individuals in the sample for the ith characteristic. Each one of the l dimensions represents a characteristic and one wants to know what are the predominant characteristics and if they bear some kind of linear relationsMathworldPlanetmath between them. This is achieved by diagonalizing the Gram matrix (often called dispersion matrix or covariance matrix in that context). The higher the eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath, the more important the eigenvectorMathworldPlanetmathPlanetmathPlanetmath associated to it.

Title Gram matrix
Canonical name GramMatrix
Date of creation 2013-03-22 17:56:55
Last modified on 2013-03-22 17:56:55
Owner lalberti (18937)
Last modified by lalberti (18937)
Numerical id 5
Author lalberti (18937)
Entry type Definition
Classification msc 15A63
Synonym Gram matrices
Synonym Gram bilinear form