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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 m × n {\displaystyle m\times n} matrix.
The singular values are non-negative real numbers, usually listed in decreasing order (σ 1 (T), σ 2 (T), …). The largest singular value σ 1 (T) is equal to the operator norm of T (see Min-max theorem). Visualization of a singular value decomposition (SVD) of a 2-dimensional, real shearing matrix M.
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD).The two versions differ because one version decomposes two matrices (somewhat like the higher-order or tensor SVD) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD.
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal ...
Latent semantic indexing (LSI) is an indexing and retrieval method that uses a mathematical technique called singular value decomposition (SVD) to identify patterns in the relationships between the terms and concepts contained in an unstructured collection of text. LSI is based on the principle that words that are used in the same contexts tend ...
Perform the singular value decomposition (SVD) ... (1982): "Examples of extended empirical orthogonal function analyses," Mon. Weather Rev., 110, 784–812.
1 Examples. Toggle Examples subsection. 1.1 Applications. 2 Properties. ... The diagonalization of the Gram matrix is the singular value decomposition. Properties
One variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one, [1] and the singular value decomposition (SVD) uses this method as well. Bidiagonalization [ edit ]