A Gram matrix is a symmetric matrix derived from the inner products of a set of vectors, often used to understand the geometry of the vectors in a given space. It plays a crucial role in kernel methods, where it helps in computing the similarity between data points in a transformed feature space without explicitly mapping them to that space.

Gram Matrix

Gram matrix

A symmetric matrix is a square matrix that is equal to its transpose, meaning the element at row i, column j is the same as the element at row j, column i. This property makes symmetric matrices particularly important in linear algebra, as they often have real eigenvalues and orthogonal eigenvectors, simplifying many mathematical computations.

symmetric matrix

inner products

set of vectors

geometry of the vectors

given space

Kernel methods are a class of algorithms for pattern analysis, where they implicitly map input data into high-dimensional feature spaces to make it easier to classify or analyze. They are particularly powerful in handling non-linear relationships by using kernel functions, which compute the inner products between the images of all pairs of data in the feature space without explicitly mapping them.

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