Singular Value Decomposition

A factorization of a finite-dimensional linear map (or matrix) A as A = U Σ V^*, where U and V are unitary (or orthogonal) matrices and Σ is a diagonal matrix with nonnegative real entries called singular values.

A factorization of an m×n matrix A (real or complex) as A = U Σ V*, where U and V are unitary (orthogonal in the real case) matrices and Σ is a diagonal (rectangular) matrix with nonnegative real entries called singular values; it encodes the action of A on orthogonal directions.

A factorization of a matrix A into A = U Σ V^* where U and V are unitary (orthogonal in real case) matrices and Σ is a diagonal (rectangular) matrix of nonnegative singular values, exposing rank, range, and nullspace structure.