Singulärvärdesdekomposition

Singulärvärdesuppdelning, often abbreviated as SVD, is a fundamental matrix factorization technique in linear algebra. It decomposes any matrix into the product of three other matrices. Specifically, for a given m x n matrix A, its singular value decomposition is expressed as A = UΣV^T. Here, U is an m x m orthogonal matrix, Σ is an m x n rectangular diagonal matrix with non-negative real numbers on its 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. These singular values are always non-negative and are typically arranged in descending order. The columns of U are called the left singular vectors, and the columns of V are called the right singular vectors. The left and right singular vectors are orthonormal.

Singular value decomposition has a wide range of applications across various fields. In image processing, it is used for image compression by retaining only the most significant singular values. In statistics and machine learning, it forms the basis for dimensionality reduction techniques like Principal Component Analysis (PCA). It is also crucial in solving linear least squares problems, recommender systems, and signal processing. The number of non-zero singular values of a matrix is equal to its rank. The SVD provides a stable and robust way to analyze and manipulate matrices, even if they are not square or invertible.