Gram matrix

A Gram matrix is the matrix of pairwise inner products among a finite collection of vectors. It is symmetric positive semidefinite and records geometric relationships such as lengths and angles.

Inner product: A scalar-valued operation on pairs of vectors that generalizes the dot product and measures their alignment. Each Gram-matrix entry is an inner product of two vectors.
Euclidean space: A real vector space equipped with the standard dot product and its associated distance. Finite collections of Euclidean vectors provide the standard setting for Gram matrices.
Kernel method: A machine-learning approach that uses pairwise evaluations of a kernel to represent similarities implicitly. Kernel matrices generalize Gram matrices to feature vectors that may never be constructed explicitly.
Distance matrix: A matrix whose entries give pairwise distances among a collection of points. It stores distances rather than inner products, though one can be reconstructed from the other under conditions.
Identity matrix: A square matrix with ones on its diagonal and zeros elsewhere. It is the Gram matrix of an orthonormal collection of vectors.
Positive-semidefinite matrix: A symmetric matrix whose quadratic form is nonnegative for every real vector. Every real Gram matrix has this property, constraining its eigenvalues and rank.
Dot product: The sum of pairwise coordinate products of two real vectors, measuring their alignment and lengths. In real coordinate space, the dot product supplies the Gram matrix entries.
Kernel matrix: A matrix whose entries are evaluations of a positive-definite kernel on pairs of data points. It is the feature-space Gram matrix central to kernel algorithms.
Covariance matrix: A matrix of pairwise covariances among the components of a random vector. It is an inner-product matrix of centered random variables, rather than a given finite vector collection.
Hilbert matrix: A matrix whose entry in row i and column j is the reciprocal of i+j−1. Its entries are inner products of monomials on the unit interval, making it a Gram matrix.

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