In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted
The diagonal entries of are uniquely determined by and are known as the singular values of . The number of non-zero singular values is equal to the rank of . The columns of and the columns of are called left-singular vectors and right-singular vectors of , respectively. They form two sets of orthonormal bases and and if they are sorted so that the singular values with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as
where is the rank of
The SVD is not unique, however it is always possible to choose the decomposition such that the singular values are in descending order. In this case, (but not and ) is uniquely determined by
The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size where is the rank of and has only the non-zero singular values. In this variant, is an semi-unitary matrix and is an semi-unitary matrix, such that
Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.
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m\times n} matrix. It is related to the polar decomposition. Specifically, the singularvaluedecomposition of an m × n {\displaystyle m\times n} complex...
rectangular diagonal matrix with the singularvalues lying on the diagonal. This is the singularvaluedecomposition. For A ∈ C m × n {\displaystyle A\in...
the generalized singularvaluedecomposition (GSVD) is the name of two different techniques based on the singularvaluedecomposition (SVD). The two versions...
the singularvaluedecomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singularvaluedecomposition. Applicable...
multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singularvaluedecomposition (SVD) of X (invented in the last quarter...
behind the construction of the polar decomposition is similar to that used to compute the singular-valuedecomposition. If A {\displaystyle A} is normal...
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of...
unique up to re-ordering. The Schmidt decomposition is essentially a restatement of the singularvaluedecomposition in a different context. Fix orthonormal...
interpretation. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singularvaluedecomposition of a covariance matrix, and...
between the singularvaluedecomposition and eigenvalue decompositions. This means that most methods for computing the singularvaluedecomposition are similar...
variation of the CP decomposition. Another popular generalization of the matrix SVD known as the higher-order singularvaluedecomposition computes orthonormal...
right-triangular system R x = Q T b . {\displaystyle Rx=Q^{T}b.} The SingularValueDecomposition (SVD) of a (tall) matrix A {\displaystyle A} is the representation...
from a large piece of text and a mathematical technique called singularvaluedecomposition (SVD) is used to reduce the number of rows while preserving the...
algebra, the complete orthogonal decomposition is a matrix decomposition. It is similar to the singularvaluedecomposition, but typically somewhat cheaper...
triangular Singularvaluedecomposition M = UΣVT, U and V orthogonal, Σ diagonal matrix Eigendecomposition of a symmetric matrix (decomposition according...
construct a full-rank factorization of A {\textstyle A} via a singularvaluedecomposition A = U Σ V ∗ = [ U 1 U 2 ] [ Σ r 0 0 0 ] [ V 1 ∗ V 2 ∗ ] = U 1...
Hermitian matrices also appear in techniques like singularvaluedecomposition (SVD) and eigenvalue decomposition. In statistics and machine learning, Hermitian...
have non-orthogonal eigenspaces. The left and right singular vectors in the singularvaluedecomposition of a normal matrix A = U Σ V ∗ {\displaystyle \mathbf...
orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called...