Singular value decomposition

The singular value decomposition factors any real or complex matrix as a product of two unitary matrices and a rectangular diagonal matrix with nonnegative entries. Its singular values quantify how the matrix stretches space along orthogonal directions.

Singular value: A nonnegative square root of an eigenvalue of a matrix multiplied by its conjugate transpose. These values form the diagonal factor and measure the matrix’s directional stretching.
Principal component analysis: A statistical method that finds orthogonal directions capturing successively less variance in data. PCA obtains its principal directions from the right singular vectors of centered data.
Matrix: A rectangular array of scalars that represents a linear map between finite-dimensional coordinate spaces. The decomposition is a structured factorization of a matrix.
Eigendecomposition: A factorization expressing a square matrix through its eigenvectors and eigenvalues when a suitable eigenbasis exists. Unlike eigendecomposition, this factorization exists for every rectangular matrix.
Left singular vector: A unit vector in the output space associated with a singular value of a matrix. Left singular vectors specify the output directions in the factorization.
Low-rank approximation: A replacement of a matrix by one of smaller rank, often chosen to minimize approximation error. Truncating the decomposition gives the best approximation of a chosen lower rank.
Linear transformation: A function between vector spaces that preserves addition and scalar multiplication. Geometrically, the decomposition describes how a linear map acts on orthogonal directions.
Polar decomposition: A factorization of a matrix into a unitary or partial isometry factor and a positive semidefinite factor. It groups the singular-vector information differently while sharing the same singular values.
Right singular vector: A unit vector in the input space associated with a singular value of a matrix. Right singular vectors specify the input directions that the matrix stretches.
Moore–Penrose inverse: A generalized inverse defined for every matrix and characterized by four algebraic conditions. Inverting nonzero singular values constructs the pseudoinverse, including for rank-deficient matrices.

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