- Gramian matrix
In

linear algebra , the**Gramian matrix**(or**Gram matrix**or**Gramian**) of a set of vectors $v\_1,dots,\; v\_n$ in aninner product space is thesymmetric matrix ofinner product s, whose entries are given by $G\_\{ij\}=(v\_i|v\_j)$.An important application is to compute

linear independence : a set of vectors is linearly independent if and only if the Gram determinant (thedeterminant of the Gram matrix) is non-zero.It is named for

Jørgen Pedersen Gram .**Examples**Most commonly, the vectors are elements of a

Euclidean space , or are functionsin an "L"^{2}space, such as continuous functions on a compact interval ["a", "b"] (which are a subspace of "L"^{ 2}( ["a", "b"] )).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:

: $G\_\{ij\}=int\_\{t\_0\}^\{t\_f\}\; ell\_i(\; au)ell\_j(\; au),\; d\; au.$

Given a matrix "A", the matrix "A"

^{T}"A" is a Gram matrix (of the columns of "A"), while the matrix "AA"^{T}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 $G\_\{i,j\}\; =\; B(v\_i,v\_j)\; ,$. The matrix will be symmetric if the bilinear form "B" is.**Applications*** If the vectors are centered

random variable s, the Gramian is the.covariance matrix

*Inquantum chemistry , the Gram matrix of a set ofbasis vectors is the.overlap matrix

*Incontrol theory (or more generallysystems theory ), theandcontrollability Gramian determine properties of a linear system.observability Gramian

* Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp. 79–94).

* In thefinite 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.**Properties****Positive semidefinite**The Gramian matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. This set of vectors is not in general unique: 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 "P"^{T}"GP".**Gram determinant**The

**Gram determinant**or**Gramian**is the determinant of the Gram matrix::$G(x\_1,dots,\; x\_n)=egin\{vmatrix\}\; (x\_1|x\_1)\; (x\_1|x\_2)\; dots\; (x\_1|x\_n)\backslash \; (x\_2|x\_1)\; (x\_2|x\_2)\; dots\; (x\_2|x\_n)\backslash vdotsvdotsvdots\backslash \; (x\_n|x\_1)\; (x\_n|x\_2)\; dots\; (x\_n|x\_n)end\{vmatrix\}.$

Geometrically, the Gram determinant is the square of the volume of the

parallelepiped 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).**External links*** cite journal

title=The Gramian and K-Volume in N-Space: Some Classical Results in Linear Algebra

last=Barth

first=Nils

url=http://www.jyi.org/volumes/volume2/issue1/articles/barth.html

journal=Journal of Young Investigators

volume=2

year=1999* " [

*http://www.owlnet.rice.edu/~fjones/chap8.pdf Volumes of parallelograms*] " by Frank Jones

*Wikimedia Foundation.
2010.*

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**Controllability Gramian**— In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time invariant linear system the controllability Gramian is given by If the matrix Wc is nonsingular, i.e. Wc… … Wikipedia**Observability Gramian**— The Observability Gramian is a Gramian used in optimal control theory to determine whether or not a linear system is observable. For a linear system described by The observability Gramian is given by If and only if the matrix Wo is nonsingular… … Wikipedia**Covariance matrix**— A bivariate Gaussian probability density function centered at (0,0), with covariance matrix [ 1.00, .50 ; .50, 1.00 ] … Wikipedia**Hilbert matrix**— In linear algebra, a Hilbert matrix is a matrix with the unit fraction elements: H {ij} = frac{1}{i+j 1}. For example, this is the 5 times; 5 Hilbert matrix::H = egin{bmatrix} 1 frac{1}{2} frac{1}{3} frac{1}{4} frac{1}{5} [4pt] frac{1}{2}… … Wikipedia**Overlap matrix**— The overlap matrix is a square matrix, used in quantum chemistry to describe the inter relationship of a set of basis vectors of a quantum system. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal.… … Wikipedia**List of matrices**— This page lists some important classes of matrices used in mathematics, science and engineering: Matrices in mathematics*(0,1) matrix a matrix with all elements either 0 or 1. Also called a binary matrix . *Adjugate matrix * Alternant matrix a… … Wikipedia**Finite element method**— The finite element method (FEM) (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based … Wikipedia**List of mathematics articles (G)**— NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… … Wikipedia**Positive definite kernel**— In operator theory, a positive definite kernel is a generalization of a positive matrix. Definition Let :{ H n } {n in {mathbb Z be a sequence of (complex) Hilbert spaces and :mathcal{L}(H i, H j)be the bounded operators from Hi to Hj . A map A… … Wikipedia**Gram (disambiguation)**— Gram is a unit of measurement of mass. Otherwise, gram may refer to: gram, the Greek based suffix meaning drawing or representation. Contents 1 Places 2 People … Wikipedia