Singular Value Decomposition
Singular Value DecompositionIt is a matrix decomposition method. The basis of eigenvector decomposition of symmetric arrays is spectral analysis, and singular value decomposition is the generalization of spectral analysis theory to arbitrary matrices.
Theoretical description
Assume that M is an m×n matrix whose elements all belong to field K, which is the real or complex field. Then there exists a decomposition such that M = UΣV*, where U is an m×m unitary matrix; Σ is an m×n non-negative real diagonal matrix; and V*, the conjugate transpose of V, is an n×n unitary matrix. Such a decomposition is called the singular value decomposition of M, and the elements Σi,i on the diagonal of Σ are the singular values of M.
In the singular value decomposition of the matrix M M = UΣV*
- The columns of V form a set of pairs of M The orthogonal "input" or "analysis" basis vectors of . These vectors are M*M The feature vector of .
- The columns of U form a set of pairs of M The basis vectors of the orthogonal "output" of . These vectors are MM* The feature vector of .
- The elements on the diagonal of Σ are singular values, which can be thought of as scalar "expansion controls" between the input and output. These are MM* and M*M The non-negative square roots of the eigenvalues of , corresponding to the row vectors of U and V .
Graphical representation and geometric meaning
Singular value decomposition can be viewed as three steps of matrix decomposition: rotating Vt, scaling Σ, and rotating U again
Singular Value Decomposition Applications
- Find the generalized inverse matrix
- Give a representation of the column space, null space, and rank of a matrix
- Finding Matrix Approximations