singularvärdesförstoring

Singular value decomposition, often abbreviated as SVD, is a factorization of a real or complex matrix. Given an m x n matrix A, its singular value decomposition is a factorization of the form A = UΣV*, where U is an m x m unitary matrix, Σ is an m x n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V* is the conjugate transpose of an n x n unitary matrix V. The diagonal entries of Σ are known as the singular values of A, and they are typically listed in descending order. The columns of U are the left singular vectors, and the columns of V are the right singular vectors.

The singular value decomposition is a powerful tool in linear algebra with numerous applications. It provides insights into the structure of a matrix, revealing its rank, null space, and column space. SVD is fundamental to principal component analysis (PCA), which is widely used for dimensionality reduction and data visualization. It also plays a crucial role in image compression, noise reduction, and recommender systems, such as those used by streaming services. In signal processing, SVD can be employed for filtering and feature extraction. The decomposition is unique up to the signs of the singular vectors and the ordering of singular values and their corresponding vectors. If all singular values are non-zero, then the matrix is invertible, and its inverse can be constructed using the SVD components.