In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} in an inner product...

systems. The Gram matrix is also important in deep learning, where it is used to represent the distribution of features in style transfer. Gram was the first...

{\displaystyle ({a_{ij}}^{s})} is a Euclidean distance matrix for every 0<s<1. The Gram matrix of a sequence of points x 1 , x 2 , … , x n {\displaystyle...

definite if and only if it is the Gram matrix of some linearly independent vectors. In general, the rank of the Gram matrix of vectors b 1 , … , b n {\displaystyle...

decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed...

matrix. (This is the rank–nullity theorem.) If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are...

congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional...

an n×n {1, −1} matrix is an integer multiple of 2n−1. Let R be an n by n {1, −1} matrix. The Gram matrix of R is defined to be the matrix G = RRT. From...

.} The matrix X T X {\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {X} } is known as the normal matrix or Gram matrix and the matrix X T y {\displaystyle...

represented in a kernel matrix (or, Gram matrix). Many algorithms can solve machine learning problems using the kernel matrix. The main problem of kernel...

used in Gaussian processes, then the Gram matrix K {\displaystyle \mathbf {K} } can also be called a covariance matrix. Application areas of kernel methods...

regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables...

potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations Dimensionality-independent rates of convergence for the empirical...

covariance matrix Wishart distribution Outer product— X X ⊤ {\displaystyle XX^{\top }} or X⊗X is the outer product of X with itself. Gram matrix Raghavan...

to the square roots of the absolute values of the eigenvalues of the Gram matrix X ∗ X {\displaystyle X^{*}X} . And if A = LU is an LU factorization of...

an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can...

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express...

eigenspace associated with an eigenvalue Gram–Schmidt process – Orthonormalization of a set of vectors Irregular matrix Matrix calculus – Specialized notation...

uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set. Gaussian function...

rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...

} The objective function can be rewritten purely in the form of the Gram matrix: T ( Y ) = 1 2 N ∑ i , j | Y i − Y j | 2 = 1 2 N ∑ i , j ( Y i 2 + Y...

formula. Metric tensor Second fundamental form Third fundamental form Tautological one-form Gram matrix First Fundamental Form — from Wolfram MathWorld...

group. The necessary and sufficient conditions for the immanant of a Gram matrix to be 0 {\displaystyle 0} are given by Gamas's Theorem. D. E. Littlewood;...

L being a lower triangular matrix. There are several methods for actually computing the QR decomposition, such as the Gram–Schmidt process, Householder...

have been centered. In PCA/SVD, we construct covariance matrix F {\displaystyle F} and Gram matrix G {\displaystyle G} F = X X T {\displaystyle F=XX^{\mathsf...

functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from...

h}} has the same matrix as the Shapovalov form S L , L ′ ( c , h ) {\displaystyle S_{L,L'}(c,h)} , now interpreted as a Gram matrix. The representation...

the package LLLBases Magma as the functions LLL and LLLGram (taking a gram matrix) Maple as the function IntegerRelations[LLL] Mathematica as the function...

{Y}}\end{bmatrix}}\in \mathbb {R} ^{n\times N}} We then compute the Gram matrix Q = X ⊤ X {\displaystyle Q=X^{\top }X} . Finally, c h {\displaystyle...

positive. The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the...