In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.
Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form
where U is an m×m real or complex unitary matrix, Σ is an m×n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and V* (the conjugate transpose of V) is an n×n real or complex unitary matrix. The diagonal entries Σi,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:
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- 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*.
Applications which employ the SVD include computing the pseudoinverse, least squares fitting of data, matrix approximation, and determining the rank, range and null space of a matrix.
Read more about Singular Value Decomposition: Statement of The Theorem, Example, Singular Values, Singular Vectors, and Their Relation To The SVD, Relation To Eigenvalue Decomposition, Existence, Geometric Meaning, Reduced SVDs, Tensor SVD, Bounded Operators On Hilbert Spaces, History
Famous quotes containing the word singular:
“It is singular how soon we lose the impression of what ceases to be constantly before us. A year impairs, a lustre obliterates. There is little distinct left without an effort of memory, then indeed the lights are rekindled for a moment—but who can be sure that the Imagination is not the torch-bearer?”
—George Gordon Noel Byron (1788–1824)