singularværdierne

Singularværdierne, also known as singular values, are a set of non-negative real numbers associated with a matrix in linear algebra. They play a crucial role in various applications, including data analysis, signal processing, and machine learning. The singular values of a matrix A are the square roots of the eigenvalues of the matrix A^T A, where A^T denotes the transpose of A. They provide a measure of the "size" or "magnitude" of the matrix, and they are invariant under orthogonal transformations, making them useful in many contexts.

The singular values of a matrix A are typically denoted by σ1, σ2, ..., σn, where n is the smaller dimension of A. The largest singular value, σ1, represents the largest possible scaling factor by which a vector can be stretched when transformed by A. The smallest singular value, σn, represents the smallest possible scaling factor. If A is a square matrix, the product of its singular values is equal to the absolute value of its determinant.

Singular values are computed using singular value decomposition (SVD), a factorization of a matrix into three other matrices. The SVD of a matrix A is given by A = UΣV^T, where U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A. The SVD is a powerful tool in linear algebra and has numerous applications in various fields.