Singular Value Decomposition the singular value decomposition of an m × n real or complex matrix M is a factorization of the form M = UΣV^{∗}, where U is an m × m real or complex unitary matrix(multiplying by their respective conjugate transposes yields identity matrices,), Σ is an m × n rectangular diagonal matrix with non-negative real numbers on the diagonal, V^{∗} (the conjugate transpose of V, or simply the transpose of V if V is real) is an n × n real or complex unitary matrix. The diagonal entries Σ_{i,i} of Σ are known as the singular values of M. The m columns of U and the n columns of V are called the left-singular vectors and right-singular vectors of M, respectively.
The singular value decomposition and the eigendecomposition are closely related. Namely: -
- The left-singular vectors of
**M**are eigenvectors of**MM**^{∗}. - The right-singular vectors of
**M**are eigenvectors of**M**^{∗}**M**. - The non-zero singular values of
**M**(found on the diagonal entries of**Σ**) are the square roots of the non-zero eigenvalues of both**M**^{∗}**M**and**MM**^{∗}.
- The left-singular vectors of
M is an m × m real square matrix with positive determinant, U, V^{∗}, and Σ are real m × m matrices as well, Σ can be regarded as a scaling matrix, and U, V^{∗} can be viewed as rotation matrices. Thus the expression UΣV^{∗} can be intuitively interpreted as a rotation, a scaling, and another rotation. The Image shows: Upper Left: The unit Disc with the two canonical unit VectorsUpper Right: Unit Disc et al. transformed with M and signular Values sigma_1 and sigma_2 indicatedLower Left: The Action of V^* on the Unit disc. This is a just Rotation.Lower Right: The Action of Sigma * V^* on the Unit disc. Sigma scales in vertically and horizontally.The this special Case the singularValues are Phi and 1/Phi where Phi is the Golden Ratio. V^* is a (counter clockwise) Rotation by an angle alpha where alpha satisfies tan(alpha) = -Phi. U is a Rotation by beta with tan(beta) = Phi-1 ## The columns of |