Egenværdidekomposition

Egenværdidekomposition, often abbreviated as EVD or SVD in English, is a fundamental concept in linear algebra and a widely used matrix factorization technique. It decomposes any given matrix into three other matrices, providing insights into the matrix's underlying structure and properties.

The decomposition of a matrix A, which can be rectangular with dimensions m x n, is expressed as A = UΣVᵀ. Here, U is an m x m orthogonal matrix, whose columns are called the left singular vectors. V is an n x n orthogonal matrix, whose columns are called the right singular vectors. Σ is an m x n rectangular diagonal matrix, where the diagonal entries are the singular values of A. These singular values are non-negative and are typically arranged in descending order.

The singular values represent the magnitude of the corresponding singular vectors and are intrinsically linked to the eigenvalues of AᵀA and AAᵀ. The singular value decomposition has numerous applications across various fields. In data analysis and machine learning, it's crucial for dimensionality reduction techniques like Principal Component Analysis (PCA). It's also used for noise reduction, image compression, recommender systems, and solving systems of linear equations. The robustness and generality of EVD make it a powerful tool for understanding and manipulating data represented by matrices.