Editing Gram matrix (section)

===Applications===
* In [[Riemannian geometry]], given an embedded <math>k</math>-dimensional [[Riemannian manifold]] <math>M\subset \mathbb{R}^n</math> and a parametrization <math>\phi: U\to M</math> for {{nowrap|<math>(x_1, \ldots, x_k)\in U\subset\mathbb{R}^k</math>,}} the volume form <math>\omega</math> on <math>M</math> induced by the embedding may be computed using the Gramian of the coordinate tangent vectors: <math display="block">\omega = \sqrt{\det G}\ dx_1 \cdots dx_k,\quad G = \left[\left\langle \frac{\partial\phi}{\partial x_i},\frac{\partial\phi}{\partial x_j}\right\rangle\right].</math> This generalizes the classical surface integral of a parametrized surface <math>\phi:U\to S\subset \mathbb{R}^3</math> for <math>(x, y)\in U\subset\mathbb{R}^2</math>: <math display="block">\int_S f\ dA = \iint_U f(\phi(x, y))\, \left|\frac{\partial\phi}{\partial x}\,{\times}\,\frac{\partial\phi}{\partial y}\right|\, dx\, dy.</math>
* If the vectors are centered [[random variable]]s, 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.&nbsp;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 function]]s are often represented as Gram matrices.<ref>{{cite journal |last1=Lanckriet |first1=G. R. G. |first2=N. |last2=Cristianini |first3=P. |last3=Bartlett |first4=L. E. |last4=Ghaoui |first5=M. I. |last5=Jordan |title=Learning the kernel matrix with semidefinite programming |journal=Journal of Machine Learning Research |volume=5 |year=2004 |pages=27–72 [p. 29] |url=https://dl.acm.org/citation.cfm?id=894170 }}</ref> (Also see [[kernel principal component analysis|kernel PCA]])
* 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]].