Ominaisarvodekompositio

Ominaisarvodekompositio, often abbreviated as SVD, is a fundamental matrix factorization technique in linear algebra. It decomposes any given matrix into a product of three specific matrices. For a real matrix A of size m x n, the SVD takes the form A = U * Sigma * V^T, where U is an m x m orthogonal matrix, Sigma is an m x n diagonal matrix, and V is an n x n orthogonal matrix. The columns of U are the left singular vectors of A, the columns of V are the right singular vectors of A, and the diagonal entries of Sigma are the singular values of A, which are always non-negative and typically ordered in descending magnitude.

The singular values represent the "strength" or importance of the corresponding singular vectors. Larger singular values indicate directions in which the matrix transformation stretches vectors the most. The SVD is a generalization of eigenvalue decomposition, which is only applicable to square matrices. However, SVD can be applied to any rectangular matrix.

This decomposition has wide-ranging applications across various fields. In data analysis and machine learning, SVD is used for dimensionality reduction through techniques like Principal Component Analysis (PCA), noise reduction by discarding smaller singular values, and recommender systems for predicting user preferences. It also plays a crucial role in image compression, solving linear least squares problems, and computing the pseudoinverse of a matrix. The existence and uniqueness of the SVD for any matrix make it a robust and powerful tool in numerical linear algebra.