Singulärvärdesuppslag

Singulärvärdesuppslag, often referred to by its English name Singular Value Decomposition (SVD), is a fundamental matrix factorization technique in linear algebra. It decomposes any given matrix into three other matrices. Specifically, for a real matrix A of size m x n, SVD expresses it as A = U * Σ * V^T, where U is an m x m orthogonal matrix, Σ (Sigma) is an m x n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V^T is the transpose of an n x n orthogonal matrix V. The diagonal entries of Σ, denoted as σ_i, are known as the singular values of A, and they are typically arranged in descending order.

The columns of U are the left singular vectors, and the columns of V are the right singular vectors. These vectors form orthonormal bases for the column space and row space of A, respectively. The singular values represent the "strength" or importance of the corresponding singular vectors. Larger singular values indicate directions where the data has more variance.

SVD has numerous applications across various fields. In data compression, it allows for the approximation of a matrix by keeping only the largest singular values and their corresponding singular vectors, effectively reducing the dimensionality of the data. It is also crucial in noise reduction, where small singular values often correspond to noise. In recommendation systems, SVD is used to identify underlying patterns in user-item interaction matrices. Furthermore, it plays a role in solving linear least squares problems, calculating matrix pseudoinverses, and in principal component analysis (PCA). The robustness and wide applicability of SVD make it a cornerstone of modern data analysis and machine learning.