singulärvärdesfaktorisering

Singulärvärdesdekomposition, also known as singular value decomposition (SVD), is a matrix factorization technique used in linear algebra and numerical analysis. It decomposes a given matrix into three other matrices, revealing the underlying structure of the original matrix. The SVD of a matrix A is typically written as A = UΣV*, where U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A.

The columns of U are called the left singular vectors, the columns of V are the right singular vectors, and the diagonal elements of Σ are the singular values. The singular values are non-negative and are typically ordered in decreasing magnitude. The number of non-zero singular values is equal to the rank of the matrix A.

SVD has numerous applications in various fields, including data compression, signal processing, and machine learning. It is used to reduce the dimensionality of data, remove noise, and extract important features. In machine learning, SVD is used in algorithms such as principal component analysis (PCA) and latent semantic analysis (LSA).

The SVD can be computed using various algorithms, such as the Golub-Kahan algorithm and the Lanczos algorithm. These algorithms are efficient and can handle large matrices. The SVD is a powerful tool in linear algebra and has wide-ranging applications in science and engineering.