SVDfaktorizáció

Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. It is a generalization of the eigendecomposition of a square matrix to any matrix. Specifically, for any real or complex matrix M of size m x n, SVD is a factorization of the form 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 M.

The singular values are the square roots of the eigenvalues of M*M or MM*. The columns of U are the left-singular vectors and are the eigenvectors of MM*. The columns of V are the right-singular vectors and are the eigenvectors of M*M.

SVD has numerous applications in various fields. It is widely used in linear algebra, numerical analysis, and data science. In signal processing and image processing, SVD can be used for dimensionality reduction, noise reduction, and image compression. In statistics and machine learning, it plays a crucial role in principal component analysis (PCA), recommender systems, and solving linear least squares problems. The ability of SVD to reveal the underlying structure of a matrix, even in the presence of noise or redundancy, makes it a powerful tool for data analysis and manipulation. Its robustness and interpretability contribute to its widespread adoption across diverse scientific and engineering disciplines.