singulärvärdena

Singulärvärdena, often translated as singular values, are fundamental components in the Singular Value Decomposition (SVD) of a matrix. SVD is a factorization of a real or complex matrix into three other matrices. For any given matrix A, its SVD takes the form A = UΣVᵀ, where U and V are orthogonal or unitary matrices, and Σ is a diagonal matrix. The entries on the diagonal of Σ are the singular values of A.

These singular values are always non-negative real numbers and are typically ordered in descending magnitude. They represent the scaling factors associated with the principal components of the data represented by the matrix. The largest singular value corresponds to the direction of greatest variance in the data, and subsequent singular values correspond to directions of decreasing variance.

The number of non-zero singular values is equal to the rank of the matrix. This property makes singular values crucial for determining the rank of a matrix, which in turn informs us about the linear independence of its rows and columns. Furthermore, singular values play a vital role in various applications, including dimensionality reduction techniques like Principal Component Analysis (PCA), image compression, noise reduction, and solving systems of linear equations. Their magnitude provides insight into the "importance" of different dimensions or components within the data.