Mátrixdekompozíció

Mátrixdekompozíció, often translated as matrix decomposition or matrix factorization, refers to the process of breaking down a given matrix into a product of matrices with specific properties. This technique is fundamental in linear algebra and has widespread applications in various fields, including data science, machine learning, and numerical analysis. The goal of matrix decomposition is often to simplify the original matrix, reveal underlying structures, or facilitate computations such as solving systems of linear equations, finding eigenvalues, or performing dimensionality reduction.

There are numerous types of matrix decomposition, each tailored for different purposes and matrix characteristics. Common examples include LU decomposition, which factors a matrix into a lower triangular matrix (L) and an upper triangular matrix (U), making it efficient for solving linear systems. QR decomposition separates a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R), useful in least squares problems and eigenvalue algorithms. Singular Value Decomposition (SVD) is a particularly powerful technique that decomposes any matrix into three other matrices, revealing important information about the matrix's rank and null space. It is extensively used in image compression, recommender systems, and noise reduction. Other notable decompositions include Cholesky decomposition for symmetric positive-definite matrices and eigenvalue decomposition for diagonalizable matrices. The choice of decomposition method depends heavily on the properties of the matrix and the specific problem being addressed.