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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the singular value decomposition (SVD) is a
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
of a real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to nor ...
with an orthonormal eigenbasis to any matrix. It is related to the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
. Specifically, the singular value decomposition of an m \times n complex matrix is a factorization of the form \mathbf = \mathbf, where is an complex
unitary matrix In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate ...
, \mathbf \Sigma is an m \times n rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \mathbf V^* is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate ...
of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
matrices; in such contexts, the SVD is often denoted \mathbf U \mathbf \Sigma \mathbf V^\mathrm. The diagonal entries \sigma_i = \Sigma_ of \mathbf \Sigma 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 In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, th ...
and and if they are sorted so that the singular values \sigma_i with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as \mathbf = \sum_^\sigma_i\mathbf_i\mathbf_i^, where r \leq \min\ is the rank of The SVD is not unique. However, it is always possible to choose the decomposition such that the singular values \Sigma_ are in descending order. In this case, \mathbf \Sigma (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 \mathbf is an semi-unitary matrix, such that \mathbf U^* \mathbf U = \mathbf V^* \mathbf V = \mathbf I_r. Mathematical applications of the SVD include computing the
pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ...
, matrix approximation, and determining the rank,
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
, and
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear ...
of a matrix. The SVD is also extremely useful in many areas of science,
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, such as
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
,
least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
fitting of data, and
process control Industrial process control (IPC) or simply process control is a system used in modern manufacturing which uses the principles of control theory and physical industrial control systems to monitor, control and optimize continuous Industrial processe ...
.


Intuitive interpretations


Rotation, coordinate scaling, and reflection

In the special case when is an real
square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...
, the matrices and can be chosen to be real matrices too. In that case, "unitary" is the same as "
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as as a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
of the space the matrices and represent rotations or reflection of the space, while represents the
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
of each coordinate by the factor Thus the SVD decomposition breaks down any linear transformation of into a
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
of three geometrical transformations: a rotation or reflection followed by a coordinate-by-coordinate
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
followed by another rotation or reflection In particular, if has a positive determinant, then and can be chosen to be both rotations with reflections, or both rotations without reflections. If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type. If the matrix is real but not square, namely with it can be interpreted as a linear transformation from to Then and can be chosen to be rotations/reflections of and respectively; and besides scaling the first coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn into


Singular values as semiaxes of an ellipse or ellipsoid

As shown in the figure, the singular values can be interpreted as the magnitude of the semiaxes of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
in 2D. This concept can be generalized to -dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, with the singular values of any
square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...
being viewed as the magnitude of the semiaxis of an -dimensional
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
. Similarly, the singular values of any matrix can be viewed as the magnitude of the semiaxis of an -dimensional
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
in -dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction. See
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...
for further details.


The columns of and are orthonormal bases

Since and are unitary, the columns of each of them form a set of orthonormal vectors, which can be regarded as basis vectors. The matrix maps the basis vector to the stretched unit vector By the definition of a unitary matrix, the same is true for their conjugate transposes and except the geometric interpretation of the singular values as stretches is lost. In short, the columns of and are
orthonormal bases In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, th ...
. When is a positive-semidefinite
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
, and are both equal to the unitary matrix used to diagonalize However, when is not positive-semidefinite and Hermitian but still
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not ...
, its eigendecomposition and singular value decomposition are distinct.


Relation to the four fundamental subspaces

* The first columns of are a basis of the
column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matr ...
of . * The last columns of are a basis of the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear ...
of . * The first columns of are a basis of the column space of (the
row space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matr ...
of in the real case). * The last columns of are a basis of the null space of .


Geometric meaning

Because and are unitary, we know that the columns of yield an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
of and the columns of yield an orthonormal basis of (with respect to the standard
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
s on these spaces). The
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
T : \left\{\begin{aligned} K^n &\to K^m \\ x &\mapsto \mathbf{M}x \end{aligned}\right. has a particularly simple description with respect to these orthonormal bases: we have T(\mathbf{V}_i) = \sigma_i \mathbf{U}_i, \qquad i = 1, \ldots, \min(m, n), where is the -th diagonal entry of and for The geometric content of the SVD theorem can thus be summarized as follows: for every linear map one can find orthonormal bases of and such that maps the -th basis vector of to a non-negative multiple of the -th basis vector of and sends the leftover basis vectors to zero. With respect to these bases, the map is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere of radius one in The linear map maps this sphere onto an
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
in Non-zero singular values are simply the lengths of the semi-axes of this ellipsoid. Especially when and all the singular values are distinct and non-zero, the SVD of the linear map can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid and specifically its axes; then consider the directions in sent by onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry sending these directions to the coordinate axes of On a second move, apply an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of as stretching coefficients. The composition then sends the unit-sphere onto an ellipsoid isometric to To define the third and last move, apply an isometry to this ellipsoid to obtain As can be easily checked, the composition coincides with


Example

Consider the matrix \mathbf{M} = \begin{bmatrix} 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \end{bmatrix} A singular value decomposition of this matrix is given by \begin{align} \mathbf{U} &= \begin{bmatrix} \color{Green}0 & \color{Blue}-1 & \color{Cyan}0 & \color{Emerald}0 \\ \color{Green}-1 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}0 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}-1 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}-1 & \color{Emerald}0 \end{bmatrix} \\ pt \mathbf \Sigma &= \begin{bmatrix} 3 & 0 & 0 & 0 & \color{Gray}\mathit{0} \\ 0 & \sqrt{5} & 0 & 0 & \color{Gray}\mathit{0} \\ 0 & 0 & 2 & 0 & \color{Gray}\mathit{0} \\ 0 & 0 & 0 & \color{Red}\mathbf{0} & \color{Gray}\mathit{0} \end{bmatrix} \\ pt \mathbf{V}^* &= \begin{bmatrix} \color{Violet}0 & \color{Violet}0 & \color{Violet}-1 & \color{Violet}0 &\color{Violet}0 \\ \color{Plum}-\sqrt{0.2}& \color{Plum}0 & \color{Plum}0 & \color{Plum}0 &\color{Plum}-\sqrt{0.8} \\ \color{Magenta}0 & \color{Magenta}-1 & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \\ \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}1 &\color{Orchid}0 \\ \color{Purple} - \sqrt{0.8} & \color{Purple}0 & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.2} \end{bmatrix} \end{align} The scaling matrix is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold, light blue bold in dark mode). Furthermore, because the matrices and are
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...
, multiplying by their respective conjugate transposes yields
identity matrices In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object ...
, as shown below. In this case, because and are real valued, each is an
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identi ...
. \begin{align} \mathbf{U} \mathbf{U}^* &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \mathbf{I}_4 \\ pt \mathbf{V} \mathbf{V}^* &= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \mathbf{I}_5 \end{align} This particular singular value decomposition is not unique. For instance, we can keep and the same, but change the last two rows of such that \mathbf{V}^* = \begin{bmatrix} \color{Violet}0 & \color{Violet}0 & \color{Violet}-1 & \color{Violet}0 &\color{Violet}0 \\ \color{Plum}-\sqrt{0.2}& \color{Plum}0 & \color{Plum}0 & \color{Plum}0 &\color{Plum}-\sqrt{0.8} \\ \color{Magenta}0 & \color{Magenta}-1 & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \\ \color{Orchid}\sqrt{0.4} & \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}\sqrt{0.5} & \color{Orchid}-\sqrt{0.1} \\ \color{Purple}-\sqrt{0.4} & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.5} & \color{Purple}\sqrt{0.1} \end{bmatrix} and get an equally valid singular value decomposition. As the matrix has rank 3, it has only 3 nonzero singular values. In taking the product , the final column of and the final two rows of are multiplied by zero, so have no effect on the matrix product, and can be replaced by any unit vectors which are orthogonal to the first three and to each-other. The compact SVD, , eliminates these superfluous rows, columns, and singular values: \begin{align} \mathbf{U}_r &= \begin{bmatrix} \color{Green}0 & \color{Blue}-1 & \color{Cyan}0 \\ \color{Green}-1 & \color{Blue}0 & \color{Cyan}0 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}0 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}-1 \end{bmatrix} \\ pt \mathbf \Sigma_r &= \begin{bmatrix} 3 & 0 & 0 \\ 0 & \sqrt{5} & 0 \\ 0 & 0 & 2 \end{bmatrix} \\ pt \mathbf{V}^*_r &= \begin{bmatrix} \color{Violet}0 & \color{Violet}0 & \color{Violet}-1 & \color{Violet}0 &\color{Violet}0 \\ \color{Plum}-\sqrt{0.2}& \color{Plum}0 & \color{Plum}0 & \color{Plum}0 &\color{Plum}-\sqrt{0.8} \\ \color{Magenta}0 & \color{Magenta}-1 & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \end{bmatrix} \end{align}


SVD and spectral decomposition


Singular values, singular vectors, and their relation to the SVD

A non-negative real number is a singular value for if and only if there exist unit-length vectors in and in such that \begin{align} \mathbf{M v} &= \sigma \mathbf{u}, \\ mu\mathbf M^*\mathbf u &= \sigma \mathbf{v}. \end{align} The vectors and are called left-singular and right-singular vectors for respectively. In any singular value decomposition \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^* the diagonal entries of are equal to the singular values of The first columns of and are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that: * An matrix has at most distinct singular values. * It is always possible to find a unitary basis for with a subset of basis vectors spanning the left-singular vectors of each singular value of * It is always possible to find a unitary basis for with a subset of basis vectors spanning the right-singular vectors of each singular value of A singular value for which we can find two left (or right) singular vectors that are linearly independent is called ''degenerate''. If and are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. The number of independent left and right-singular vectors coincides, and these singular vectors appear in the same columns of and corresponding to diagonal elements of all with the same value As an exception, the left and right-singular vectors of singular value 0 comprise all unit vectors in the
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
and kernel, respectively, of which by the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts: * the number of columns of a matrix is the sum of the rank of and the nullity of ; and * the dimension of the domain of a linear transformation is the sum of the r ...
cannot be the same dimension if Even if all singular values are nonzero, if then the cokernel is nontrivial, in which case is padded with orthogonal vectors from the cokernel. Conversely, if then is padded by orthogonal vectors from the kernel. However, if the singular value of exists, the extra columns of or already appear as left or right-singular vectors. Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor (for the real case up to a sign). Consequently, if all singular values of a square matrix are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of by a unit-phase factor and simultaneous multiplication of the corresponding column of by the same unit-phase factor. In general, the SVD is unique up to arbitrary unitary transformations applied uniformly to the column vectors of both and spanning the subspaces of each singular value, and up to arbitrary unitary transformations on vectors of and spanning the kernel and cokernel, respectively, of


Relation to eigenvalue decomposition

The singular value decomposition is very general in the sense that it can be applied to any matrix, whereas eigenvalue decomposition can only be applied to square diagonalizable matrices. Nevertheless, the two decompositions are related. If has SVD the following two relations hold: \begin{align} \mathbf{M}^* \mathbf{M} &= \mathbf{V} \mathbf \Sigma^* \mathbf{U}^*\, \mathbf{U} \mathbf \Sigma \mathbf{V}^* = \mathbf{V} (\mathbf \Sigma^* \mathbf \Sigma) \mathbf{V}^*, \\ mu\mathbf{M} \mathbf{M}^* &= \mathbf{U} \mathbf \Sigma \mathbf{V}^*\, \mathbf{V} \mathbf \Sigma^* \mathbf{U}^* = \mathbf{U} (\mathbf \Sigma \mathbf \Sigma^*) \mathbf{U}^*. \end{align} The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently: * The columns of (referred to as right-singular vectors) are
eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
of * The columns of (referred to as left-singular vectors) are eigenvectors of * The non-zero elements of (non-zero singular values) are the square roots of the non-zero
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of or In the special case of being a
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to nor ...
, and thus also square, the
spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...
ensures that it can be unitarily diagonalized using a basis of
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s, and thus decomposed as for some unitary matrix and diagonal matrix with complex elements along the diagonal. When is positive semi-definite, will be non-negative real numbers so that the decomposition is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase of each to either its corresponding or The natural connection of the SVD to non-normal matrices is through the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
theorem: where is positive semidefinite and normal, and is unitary. Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of while related, differ: the eigenvalue decomposition is where is not necessarily unitary and is not necessarily positive semi-definite, while the SVD is where is diagonal and positive semi-definite, and and are unitary matrices that are not necessarily related except through the matrix While only non-defective square matrices have an eigenvalue decomposition, any matrix has a SVD.


Applications of the SVD


Pseudoinverse

The singular value decomposition can be used for computing the
pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ...
of a matrix. The pseudoinverse of the matrix with singular value decomposition is \mathbf M^+ = \mathbf V \boldsymbol \Sigma^+ \mathbf U^\ast, where \boldsymbol \Sigma^+ is the pseudoinverse of \boldsymbol \Sigma, which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The pseudoinverse is one way to solve
linear least squares Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and ...
problems.


Solving homogeneous linear equations

A set of homogeneous linear equations can be written as for a matrix and vector A typical situation is that is known and a non-zero is to be determined which satisfies the equation. Such an belongs to 's
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear ...
and is sometimes called a (right) null vector of The vector can be characterized as a right-singular vector corresponding to a singular value of that is zero. This observation means that if is a
square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...
and has no vanishing singular value, the equation has no non-zero as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero satisfying with denoting the conjugate transpose of is called a left null vector of


Total least squares minimization

A
total least squares In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generaliz ...
problem seeks the vector that minimizes the
2-norm In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and ze ...
of a vector under the constraint \, \mathbf x \, = 1. The solution turns out to be the right-singular vector of corresponding to the smallest singular value.


Range, null space and rank

Another application of the SVD is that it provides an explicit representation of the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
and
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear ...
of a matrix The right-singular vectors corresponding to vanishing singular values of span the null space of and the left-singular vectors corresponding to the non-zero singular values of span the range of For example, in the above
example Example may refer to: * ''exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, an ...
the null space is spanned by the last row of and the range is spanned by the first three columns of As a consequence, the rank of equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in \mathbf \Sigma. In numerical linear algebra the singular values can be used to determine the ''effective rank'' of a matrix, as
rounding error In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Roun ...
may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.


Low-rank matrix approximation

Some practical applications need to solve the problem of approximating a matrix with another matrix \tilde{\mathbf{M, said to be truncated, which has a specific rank . In the case that the approximation is based on minimizing the
Frobenius norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
of the difference between and under the constraint that \operatorname{rank}\bigl(\tilde{\mathbf{M\bigr) = r, it turns out that the solution is given by the SVD of namely \tilde{\mathbf{M = \mathbf{U} \tilde{\mathbf \Sigma} \mathbf{V}^*, where \tilde{\mathbf \Sigma} is the same matrix as \mathbf \Sigma except that it contains only the largest singular values (the other singular values are replaced by zero). This is known as the Eckart–Young theorem, as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see ).


Image compression

One practical consequence of the low-rank approximation given by SVD is that a greyscale image represented as an m \times n matrix \mathbf{A}, can be efficiently represented by keeping the first k singular values and corresponding vectors. The truncated decomposition \mathbf{A}_k = \sum_{j=1}^k \sigma_j\mathbf{u}_j \mathbf{v}_j^T gives an image with the best 2-norm error out of all rank k approximations. Thus, the task becomes finding an approximation that balances retaining perceptual fidelity with the number of vectors required to reconstruct the image. Storing \mathbf{A}_k requires only k(n + m + 1) floating-point numbers compared to nm integers. This same idea extends to color images by applying this operation to each channel or stacking the channels into one matrix. Since the singular values of most natural images decay quickly, most of their variance is often captured by a small k. For a 1528 × 1225 greyscale image, we can achieve a relative error of .7% with as little as k = 100. In practice, however, computing the SVD can be too computationally expensive and the resulting compression is typically less storage efficient than a specialized algorithm such as
JPEG JPEG ( , short for Joint Photographic Experts Group and sometimes retroactively referred to as JPEG 1) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degr ...
.


Separable models

The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix can be written as an
outer product In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
of two vectors or, in coordinates, Specifically, the matrix can be decomposed as, \mathbf{M} = \sum_i \mathbf{A}_i = \sum_i \sigma_i \mathbf U_i \otimes \mathbf V_i. Here and are the -th columns of the corresponding SVD matrices, are the ordered singular values, and each is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero is exactly the rank of the matrix. Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described by a
Gabor filter In image processing, a Gabor filter, named after Dennis Gabor, who first proposed it as a 1D filter. The Gabor filter was first generalized to 2D by Gösta Granlund, by adding a reference direction. The Gabor filter is a linear filter used for ...
in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example, reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two-dimensional filter (space, time) which can be decomposed through SVD. The first column of in the SVD factorization is then a Gabor while the first column of represents the time modulation (or vice versa). One may then define an index of separability \alpha = \frac{\sigma_1^2}{\sum_i \sigma_i^2}, which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.


Nearest orthogonal matrix

It is possible to use the SVD of a square matrix to determine the
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identi ...
closest to The closeness of fit is measured by the
Frobenius norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
of The solution is the product This intuitively makes sense because an orthogonal matrix would have the decomposition where is the identity matrix, so that if then the product amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix of the Polar Decomposition \mathbf M = \mathbf R \mathbf P = \mathbf P' \mathbf R in either order of stretch and rotation, as described above. A similar problem, with interesting applications in shape analysis, is the orthogonal Procrustes problem, which consists of finding an orthogonal matrix which most closely maps to Specifically, \mathbf{O} = \underset\Omega\operatorname{argmin} \, \mathbf{A}\boldsymbol{\Omega} - \mathbf{B}\, _F \quad\text{subject to}\quad \boldsymbol{\Omega}^\operatorname{T}\boldsymbol{\Omega} = \mathbf{I}, where \, \cdot \, _F denotes the Frobenius norm. This problem is equivalent to finding the nearest orthogonal matrix to a given matrix \mathbf M = \mathbf A^\operatorname{T} \mathbf B.


The Kabsch algorithm

The Kabsch algorithm (called Wahba's problem in other fields) uses SVD to compute the optimal rotation (with respect to least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.


Principal Component Analysis

The SVD can be used to construct the principal components in
principal component analysis Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that th ...
as follows: Let \mathbf{X} \in \mathbb{R}^{N \times p} be a data matrix where each of the N rows is a (feature-wise) mean-centered observation, each of dimension p. The SVD of \mathbf{X} is: \mathbf{X} = \mathbf{V} \boldsymbol{\Sigma} \mathbf{U}^\ast From the same reference, we see that \mathbf{V} \boldsymbol{\Sigma} contains the scores of the rows of \mathbf{X} (i.e. each observation), and \mathbf{U} is the matrix whose columns are principal component loading vectors.


Signal processing

The SVD and pseudoinverse have been successfully applied to
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
,
image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...
and
big data Big data primarily refers to data sets that are too large or complex to be dealt with by traditional data processing, data-processing application software, software. Data with many entries (rows) offer greater statistical power, while data with ...
(e.g., in genomic signal processing).


Other examples

The SVD is also applied extensively to the study of linear
inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in ac ...
s and is useful in the analysis of regularization methods such as that of Tikhonov. It is widely used in statistics, where it is related to
principal component analysis Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that th ...
and to
correspondence analysis Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical ...
, and in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
and
pattern recognition Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their p ...
. It is also used in output-only
modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. It consists of mechanically exciting a studied component in such a way to target the Normal mode, modeshapes of the structure, and recording the vibration ...
, where the non-scaled mode shapes can be determined from the singular vectors. Yet another usage is
latent semantic indexing Latent semantic analysis (LSA) is a technique in natural language processing, in particular distributional semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the d ...
in natural-language text processing. In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" \kappa := \sigma_\text{max} / \sigma_\text{min}. It often controls the error rate or convergence rate of a given computational scheme on such systems. The SVD also plays a crucial role in the field of
quantum information Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
, in a form often referred to as the
Schmidt decomposition In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information ...
. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be entangled: if the rank of the \mathbf \Sigma matrix is larger than one. One application of SVD to rather large matrices is in
numerical weather prediction Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to weather forecasting, predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of comput ...
, where Lanczos methods are used to estimate the most linearly quickly growing few perturbations to the central numerical weather prediction over a given initial forward time period; i.e., the singular vectors corresponding to the largest singular values of the linearized propagator for the global weather over that time interval. The output singular vectors in this case are entire weather systems. These perturbations are then run through the full nonlinear model to generate an ensemble forecast, giving a handle on some of the uncertainty that should be allowed for around the current central prediction. SVD has also been applied to reduced order modelling. The aim of reduced order modelling is to reduce the number of degrees of freedom in a complex system which is to be modeled. SVD was coupled with radial basis functions to interpolate solutions to three-dimensional unsteady flow problems. Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO. SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models. Singular value decomposition is used in
recommender systems A recommender system (RecSys), or a recommendation system (sometimes replacing ''system'' with terms such as ''platform'', ''engine'', or ''algorithm'') and sometimes only called "the algorithm" or "algorithm", is a subclass of information fil ...
to predict people's item ratings. Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines. Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease
outbreak In epidemiology, an outbreak is a sudden increase in occurrences of a disease when cases are in excess of normal expectancy for the location or season. It may affect a small and localized group or impact upon thousands of people across an entire ...
detection. A combination of SVD and higher-order SVD also has been applied for real time event detection from complex data streams (multivariate data with space and time dimensions) in
disease surveillance Disease surveillance is an epidemiological practice by which the spread of disease is monitored in order to establish patterns of progression. The main role of disease surveillance is to predict, observe, and minimize the harm caused by outbrea ...
. In
astrodynamics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of univ ...
, the SVD and its variants are used as an option to determine suitable maneuver directions for transfer trajectory design and
orbital station-keeping In astrodynamics, orbital station-keeping is keeping a spacecraft at a fixed distance from another spacecraft or celestial body. It requires a series of orbital maneuvers made with thruster burns to keep the active craft in the same orbit as its ...
. The SVD can be used to measure the similarity between real-valued matrices. By measuring the angles between the singular vectors, the inherent two-dimensional structure of matrices is accounted for. This method was shown to outperform
cosine similarity In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided ...
and
Frobenius norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
in most cases, including brain activity measurements from
neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, ...
experiments.


Proof of existence

An eigenvalue of a matrix is characterized by the algebraic relation When is
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
, a variational characterization is also available. Let be a real
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
. Define f : \left\{ \begin{align} \R^n &\to \R \\ \mathbf{x} &\mapsto \mathbf{x}^\operatorname{T} \mathbf{M} \mathbf{x} \end{align}\right. By the
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed and bounded interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and ...
, this continuous function attains a maximum at some when restricted to the unit sphere \{\, \mathbf x\, = 1\}. By the
Lagrange multipliers In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
theorem, necessarily satisfies \nabla \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{u} - \lambda \cdot \nabla \mathbf{u}^\operatorname{T} \mathbf{u} = 0 for some real number The nabla symbol, , is the
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
operator (differentiation with respect to Using the symmetry of we obtain \nabla \mathbf{x}^\operatorname{T} \mathbf{M} \mathbf{x} - \lambda \cdot \nabla \mathbf{x}^\operatorname{T} \mathbf{x} = 2(\mathbf{M}-\lambda \mathbf{I})\mathbf{x}. Therefore so is a unit length eigenvector of For every unit length eigenvector of its eigenvalue is so is the largest eigenvalue of The same calculation performed on the orthogonal complement of gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there is a real-valued function of real variables. Singular values are similar in that they can be described algebraically or from variational principles. Although, unlike the eigenvalue case, Hermiticity, or symmetry, of is no longer required. This section gives these two arguments for existence of singular value decomposition.


Based on the spectral theorem

Let \mathbf{M} be an complex matrix. Since \mathbf{M}^* \mathbf{M} is positive semi-definite and Hermitian, by the
spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...
, there exists an unitary matrix \mathbf{V} such that \mathbf V^* \mathbf M^* \mathbf M \mathbf V = \bar\mathbf{D} = \begin{bmatrix} \mathbf{D} & 0 \\ 0 & 0\end{bmatrix}, where \mathbf{D} is diagonal and positive definite, of dimension \ell\times \ell, with \ell the number of non-zero eigenvalues of \mathbf{M}^* \mathbf{M} (which can be shown to verify \ell\le\min(n,m)). Note that \mathbf{V} is here by definition a matrix whose i-th column is the i-th eigenvector of \mathbf{M}^* \mathbf{M}, corresponding to the eigenvalue \bar{\mathbf{D_{ii}. Moreover, the j-th column of \mathbf{V}, for j>\ell, is an eigenvector of \mathbf{M}^* \mathbf{M} with eigenvalue \bar{\mathbf{D_{jj}=0. This can be expressed by writing \mathbf{V} as \mathbf{V}=\begin{bmatrix}\mathbf{V}_1 &\mathbf{V}_2\end{bmatrix}, where the columns of \mathbf{V}_1 and \mathbf{V}_2 therefore contain the eigenvectors of \mathbf{M}^* \mathbf{M} corresponding to non-zero and zero eigenvalues, respectively. Using this rewriting of \mathbf{V}, the equation becomes: \begin{bmatrix} \mathbf{V}_1^* \\ \mathbf{V}_2^* \end{bmatrix} \mathbf{M}^* \mathbf{M}\, \begin{bmatrix} \mathbf{V}_1 & \!\! \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 & \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 \\ \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 & \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{D} & 0 \\ 0 & 0 \end{bmatrix}. This implies that \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 = \mathbf{D}, \quad \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 = \mathbf{0}. Moreover, the second equation implies \mathbf{M}\mathbf{V}_2 = \mathbf{0}. Finally, the unitary-ness of \mathbf{V} translates, in terms of \mathbf{V}_1 and \mathbf{V}_2, into the following conditions: \begin{align} \mathbf{V}_1^* \mathbf{V}_1 &= \mathbf{I}_1, \\ \mathbf{V}_2^* \mathbf{V}_2 &= \mathbf{I}_2, \\ \mathbf{V}_1 \mathbf{V}_1^* + \mathbf{V}_2 \mathbf{V}_2^* &= \mathbf{I}_{12}, \end{align} where the subscripts on the identity matrices are used to remark that they are of different dimensions. Let us now define \mathbf{U}_1 = \mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2. Then, \mathbf{U}_1 \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2 \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M} (\mathbf{I} - \mathbf{V}_2\mathbf{V}_2^*) = \mathbf{M} - (\mathbf{M}\mathbf{V}_2)\mathbf{V}_2^* = \mathbf{M}, since \mathbf{M}\mathbf{V}_2 = \mathbf{0}. This can be also seen as immediate consequence of the fact that \mathbf{M}\mathbf{V}_1\mathbf{V}_1^* = \mathbf{M}. This is equivalent to the observation that if \{\boldsymbol v_i\}_{i=1}^\ell is the set of eigenvectors of \mathbf{M}^* \mathbf{M} corresponding to non-vanishing eigenvalues \{\lambda_i\}_{i=1}^\ell, then \{\mathbf M \boldsymbol v_i\}_{i=1}^\ell is a set of orthogonal vectors, and \bigl\{\lambda_i^{-1/2}\mathbf M \boldsymbol v_i\bigr\}\vphantom, _{i=1}^\ell is a (generally not complete) set of ''orthonormal'' vectors. This matches with the matrix formalism used above denoting with \mathbf{V}_1 the matrix whose columns are \{\boldsymbol v_i\}_{i=1}^\ell, with \mathbf{V}_2 the matrix whose columns are the eigenvectors of \mathbf{M}^* \mathbf{M} with vanishing eigenvalue, and \mathbf{U}_1 the matrix whose columns are the vectors \bigl\{\lambda_i^{-1/2}\mathbf M \boldsymbol v_i\bigr\}\vphantom, _{i=1}^\ell. We see that this is almost the desired result, except that \mathbf{U}_1 and \mathbf{V}_1 are in general not unitary, since they might not be square. However, we do know that the number of rows of \mathbf{U}_1 is no smaller than the number of columns, since the dimensions of \mathbf{D} is no greater than m and n. Also, since \mathbf{U}_1^*\mathbf{U}_1 = \mathbf{D}^{-\frac{1}{2\mathbf{V}_1^*\mathbf{M}^*\mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2=\mathbf{D}^{-\frac{1}{2\mathbf{D}\mathbf{D}^{-\frac{1}{2 = \mathbf{I_1}, the columns in \mathbf{U}_1 are orthonormal and can be extended to an orthonormal basis. This means that we can choose \mathbf{U}_2 such that \mathbf{U} = \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix} is unitary. For we already have to make it unitary. Now, define \mathbf \Sigma = \begin{bmatrix} \begin{bmatrix} \mathbf{D}^\frac{1}{2} & 0 \\ 0 & 0 \end{bmatrix} \\ 0 \end{bmatrix}, where extra zero rows are added or removed to make the number of zero rows equal the number of columns of and hence the overall dimensions of \mathbf \Sigma equal to m\times n. Then \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix} \begin{bmatrix} \begin{bmatrix} \mathbf{}D^\frac{1}{2} & 0 \\ 0 & 0 \end{bmatrix} \\ 0 \end{bmatrix} \begin{bmatrix} \mathbf{V}_1 & \mathbf{V}_2 \end{bmatrix}^* = \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix} \begin{bmatrix} \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* \\ 0 \end{bmatrix} = \mathbf{U}_1 \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M}, which is the desired result: \mathbf{M} = \mathbf{U} \mathbf \Sigma \mathbf{V}^*. Notice the argument could begin with diagonalizing rather than (This shows directly that and have the same non-zero eigenvalues).


Based on variational characterization

The singular values can also be characterized as the maxima of considered as a function of and over particular subspaces. The singular vectors are the values of and where these maxima are attained. Let denote an matrix with real entries. Let be the unit (k-1)-sphere in \mathbb{R}^k , and define \sigma(\mathbf{u}, \mathbf{v}) = \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v}, \mathbf{u} \in S^{m-1}, \mathbf{v} \in S^{n-1}. Consider the function restricted to Since both and are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
sets, their product is also compact. Furthermore, since is continuous, it attains a largest value for at least one pair of vectors in and in This largest value is denoted and the corresponding vectors are denoted and Since is the largest value of it must be non-negative. If it were negative, changing the sign of either or would make it positive and therefore larger. Statement. and are left and right-singular vectors of with corresponding singular value Proof. Similar to the eigenvalues case, by assumption the two vectors satisfy the Lagrange multiplier equation: \nabla \sigma = \nabla \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v} - \lambda_1 \cdot \nabla \mathbf{u}^\operatorname{T} \mathbf{u} - \lambda_2 \cdot \nabla \mathbf{v}^\operatorname{T} \mathbf{v} After some algebra, this becomes \begin{align} \mathbf{M} \mathbf{v}_1 &= 2 \lambda_1 \mathbf{u}_1 + 0, \\ \mathbf{M}^\operatorname{T} \mathbf{u}_1 &= 0 + 2 \lambda_2 \mathbf{v}_1. \end{align} Multiplying the first equation from left by and the second equation from left by and taking \, \mathbf u \, = \, \mathbf v \, = 1 into account gives \sigma_1 = 2\lambda_1 = 2\lambda_2. Plugging this into the pair of equations above, we have \begin{align} \mathbf{M} \mathbf{v}_1 &= \sigma_1 \mathbf{u}_1, \\ \mathbf{M}^\operatorname{T} \mathbf{u}_1 &= \sigma_1 \mathbf{v}_1. \end{align} This proves the statement. More singular vectors and singular values can be found by maximizing over normalized and which are orthogonal to and respectively. The passage from real to complex is similar to the eigenvalue case.


Calculating the SVD


One-sided Jacobi algorithm

One-sided Jacobi algorithm is an iterative algorithm, where a matrix is iteratively transformed into a matrix with orthogonal columns. The elementary iteration is given as a Jacobi rotation, M\leftarrow MJ(p, q, \theta), where the angle \theta of the Jacobi rotation matrix J(p,q,\theta) is chosen such that after the rotation the columns with numbers p and q become orthogonal. The indices (p,q) are swept cyclically, (p=1\dots m,q=p+1\dots m), where m is the number of columns. After the algorithm has converged, the singular value decomposition M=USV^T is recovered as follows: the matrix V is the accumulation of Jacobi rotation matrices, the matrix U is given by normalising the columns of the transformed matrix M, and the singular values are given as the norms of the columns of the transformed matrix M.


Two-sided Jacobi algorithm

Two-sided Jacobi SVD algorithm—a generalization of the Jacobi eigenvalue algorithm—is an iterative algorithm where a square matrix is iteratively transformed into a diagonal matrix. If the matrix is not square the
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
is performed first and then the algorithm is applied to the R matrix. The elementary iteration zeroes a pair of off-diagonal elements by first applying a Givens rotation to symmetrize the pair of elements and then applying a Jacobi transformation to zero them, M \leftarrow J^TGMJ where G is the Givens rotation matrix with the angle chosen such that the given pair of off-diagonal elements become equal after the rotation, and where J is the Jacobi transformation matrix that zeroes these off-diagonal elements. The iterations proceeds exactly as in the Jacobi eigenvalue algorithm: by cyclic sweeps over all off-diagonal elements. After the algorithm has converged the resulting diagonal matrix contains the singular values. The matrices U and V are accumulated as follows: U\leftarrow UG^TJ, V\leftarrow VJ.


Numerical approach

The singular value decomposition can be computed using the following observations: * The left-singular vectors of are a set of
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpe ...
eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
of . * The right-singular vectors of are a set of orthonormal eigenvectors of . * The non-zero singular values of (found on the diagonal entries of \mathbf \Sigma) are the square roots of the non-zero
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of both and . The SVD of a matrix is typically computed by a two-step procedure. In the first step, the matrix is reduced to a bidiagonal matrix. This takes
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
floating-point operations (flop), assuming that The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an
iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...
(as with
eigenvalue algorithm In numerical analysis, one of the most important problems is designing efficient and Numerical stability, stable algorithms for finding the eigenvalues of a Matrix (mathematics), matrix. These eigenvalue algorithms may also find eigenvectors. Eig ...
s). However, in practice it suffices to compute the SVD up to a certain precision, like the machine epsilon. If this precision is considered constant, then the second step takes iterations, each costing flops. Thus, the first step is more expensive, and the overall cost is flops . The first step can be done using Householder reflections for a cost of flops, assuming that only the singular values are needed and not the singular vectors. If is much larger than then it is advantageous to first reduce the matrix to a triangular matrix with the
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
and then use Householder reflections to further reduce the matrix to bidiagonal form; the combined cost is flops . The second step can be done by a variant of the QR algorithm for the computation of eigenvalues, which was first described by . The LAPACK subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small . Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition. The same algorithm is implemented in the
GNU Scientific Library The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C (programming language), C; wrappers are available for other programming languages. The GSL is part of ...
(GSL). The GSL also offers an alternative method that uses a one-sided Jacobi orthogonalization in step 2 . This method computes the SVD of the bidiagonal matrix by solving a sequence of SVD problems, similar to how the Jacobi eigenvalue algorithm solves a sequence of eigenvalue methods . Yet another method for step 2 uses the idea of
divide-and-conquer eigenvalue algorithm The term divide and conquer in politics refers to an entity gaining and maintaining political power by using divisive measures. This includes the exploitation of existing divisions within a political group by its political opponents, and also ...
s . There is an alternative way that does not explicitly use the eigenvalue decomposition. Usually the singular value problem of a matrix is converted into an equivalent symmetric eigenvalue problem such as or \begin{bmatrix} \mathbf{0} & \mathbf{M} \\ \mathbf{M}^* & \mathbf{0} \end{bmatrix}. The approaches that use eigenvalue decompositions are based on the QR algorithm, which is well-developed to be stable and fast. Note that the singular values are real and right- and left- singular vectors are not required to form similarity transformations. One can iteratively alternate between the
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
and the LQ decomposition to find the real diagonal Hermitian matrices. The
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
gives and the LQ decomposition of gives Thus, at every iteration, we have update and repeat the orthogonalizations. Eventually, this iteration between
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
and LQ decomposition produces left- and right- unitary singular matrices. This approach cannot readily be accelerated, as the QR algorithm can with spectral shifts or deflation. This is because the shift method is not easily defined without using similarity transformations. However, this iterative approach is very simple to implement, so is a good choice when speed does not matter. This method also provides insight into how purely orthogonal/unitary transformations can obtain the SVD.


Analytic result of 2 × 2 SVD

The singular values of a matrix can be found analytically. Let the matrix be \mathbf{M} = z_0\mathbf{I} + z_1\sigma_1 + z_2\sigma_2 + z_3\sigma_3 where z_i \in \mathbb{C} are complex numbers that parameterize the matrix, is the identity matrix, and \sigma_i denote the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
. Then its two singular values are given by \begin{align} \sigma_\pm &= \sqrt{, z_0, ^2 + , z_1, ^2 + , z_2, ^2 + , z_3, ^2 \pm \sqrt{\bigl(, z_0, ^2 + , z_1, ^2 + , z_2, ^2 + , z_3, ^2\bigr)^2 - , z_0^2 - z_1^2 - z_2^2 - z_3^2, ^2 \\ &= \sqrt{, z_0, ^2 + , z_1, ^2 + , z_2, ^2 + , z_3, ^2 \pm 2\sqrt{(\operatorname{Re}z_0z_1^*)^2 + (\operatorname{Re}z_0z_2^*)^2 + (\operatorname{Re}z_0z_3^*)^2 + (\operatorname{Im}z_1z_2^*)^2 + (\operatorname{Im}z_2z_3^*)^2 + (\operatorname{Im}z_3z_1^*)^2 \end{align}


Reduced SVDs

In applications it is quite unusual for the full SVD, including a full unitary decomposition of the null-space of the matrix, to be required. Instead, it is often sufficient (as well as faster, and more economical for storage) to compute a reduced version of the SVD. The following can be distinguished for an matrix of rank :


Thin SVD

The thin, or economy-sized, SVD of a matrix is given by \mathbf{M} = \mathbf{U}_k \mathbf \Sigma_k \mathbf{V}^*_k, where k = \min(m, n), the matrices and contain only the first columns of and and contains only the first singular values from The matrix is thus is diagonal, and is The thin SVD uses significantly less space and computation time if The first stage in its calculation will usually be a
QR decomposition 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 an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decom ...
of which can make for a significantly quicker calculation in this case.


Compact SVD

The compact SVD of a matrix is given by \mathbf{M} = \mathbf U_r \mathbf \Sigma_r \mathbf V_r^*. Only the column vectors of and row vectors of corresponding to the non-zero singular values are calculated. The remaining vectors of and are not calculated. This is quicker and more economical than the thin SVD if The matrix is thus is diagonal, and is


Truncated SVD

In many applications the number of the non-zero singular values is large making even the Compact SVD impractical to compute. In such cases, the smallest singular values may need to be truncated to compute only non-zero singular values. The truncated SVD is no longer an exact decomposition of the original matrix but rather provides the optimal low-rank matrix approximation by any matrix of a fixed rank \tilde{\mathbf{M = \mathbf{U}_t \mathbf \Sigma_t \mathbf{V}_t^*, where matrix is is diagonal, and is Only the column vectors of and row vectors of corresponding to the largest singular values are calculated. This can be much quicker and more economical than the compact SVD if but requires a completely different toolset of numerical solvers. In applications that require an approximation to the
Moore–Penrose inverse In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Ar ...
of the matrix the smallest singular values of are of interest, which are more challenging to compute compared to the largest ones. Truncated SVD is employed in
latent semantic indexing Latent semantic analysis (LSA) is a technique in natural language processing, in particular distributional semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the d ...
.


Norms


Ky Fan norms

The sum of the largest singular values of is a
matrix norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
, the Ky Fan -norm of The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Inform ...
of as a linear operator with respect to the Euclidean norms of and In other words, the Ky Fan 1-norm is the operator norm induced by the standard \ell^2 Euclidean inner product. For this reason, it is also called the operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator on (possibly infinite-dimensional) Hilbert spaces \, \mathbf M \, = \, \mathbf M^* \mathbf M \, ^\frac{1}{2} But, in the matrix case, is a
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to nor ...
, so \, \mathbf M^* \mathbf M\, ^{1/2} is the largest eigenvalue of i.e. the largest singular value of The last of the Ky Fan norms, the sum of all singular values, is the trace norm (also known as the 'nuclear norm'), defined by \, \mathbf M \, = \operatorname{Tr}(\mathbf M^* \mathbf M)^{1/2} (the eigenvalues of are the squares of the singular values).


Hilbert–Schmidt norm

The singular values are related to another norm on the space of operators. Consider the Hilbert–Schmidt inner product on the matrices, defined by \langle \mathbf{M}, \mathbf{N} \rangle = \operatorname{tr} \left( \mathbf{N}^*\mathbf{M} \right). So the induced norm is \, \mathbf{M} \, = \sqrt{\langle \mathbf{M}, \mathbf{M} \rangle} = \sqrt{\operatorname{tr} \left( \mathbf{M}^*\mathbf{M} \right)}. Since the trace is invariant under unitary equivalence, this shows \, \mathbf{M} \, = \sqrt{\vphantom\bigg, \sum_i \sigma_i ^2} where are the singular values of This is called the
Frobenius norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
, Schatten 2-norm, or Hilbert–Schmidt norm of Direct calculation shows that the Frobenius norm of coincides with: \sqrt{\vphantom\bigg, \sum_{ij} , m_{ij} , ^2}. In addition, the Frobenius norm and the trace norm (the nuclear norm) are special cases of the Schatten norm.


Variations and generalizations


Scale-invariant SVD

The singular values of a matrix are uniquely defined and are invariant with respect to left and/or right unitary transformations of In other words, the singular values of for unitary matrices and are equal to the singular values of This is an important property for applications in which it is necessary to preserve Euclidean distances and invariance with respect to rotations. The Scale-Invariant SVD, or SI-SVD, is analogous to the conventional SVD except that its uniquely-determined singular values are invariant with respect to diagonal transformations of In other words, the singular values of for invertible diagonal matrices and are equal to the singular values of This is an important property for applications for which invariance to the choice of units on variables (e.g., metric versus imperial units) is needed.


Bounded operators on Hilbert spaces

The factorization can be extended to a
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
on a separable Hilbert space Namely, for any bounded operator there exist a
partial isometry Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
a unitary a measure space and a non-negative measurable such that \mathbf{M} = \mathbf{U} T_f \mathbf{V}^* where is the multiplication by on This can be shown by mimicking the linear algebraic argument for the matrix case above. is the unique positive square root of as given by the Borel functional calculus for
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s. The reason why need not be unitary is that, unlike the finite-dimensional case, given an isometry with nontrivial kernel, a suitable may not be found such that \begin{bmatrix} U_1 \\ U_2 \end{bmatrix} is a unitary operator. As for matrices, the singular value factorization is equivalent to the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
for operators: we can simply write \mathbf M = \mathbf U \mathbf V^* \cdot \mathbf V T_f \mathbf V^* and notice that is still a partial isometry while is positive.


Singular values and compact operators

The notion of singular values and left/right-singular vectors can be extended to compact operator on Hilbert space as they have a discrete spectrum. If is compact, every non-zero in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If is compact, so is . Applying the diagonalization result, the unitary image of its positive square root has a set of orthonormal eigenvectors corresponding to strictly positive eigenvalues . For any in \mathbf{M} \psi = \mathbf{U} T_f \mathbf{V}^* \psi = \sum_i \left \langle \mathbf{U} T_f \mathbf{V}^* \psi, \mathbf{U} e_i \right \rangle \mathbf{U} e_i = \sum_i \sigma_i \left \langle \psi, \mathbf{V} e_i \right \rangle \mathbf{U} e_i, where the series converges in the norm topology on Notice how this resembles the expression from the finite-dimensional case. are called the singular values of (resp. ) can be considered the left-singular (resp. right-singular) vectors of Compact operators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is: :Theorem. is compact if and only if is compact.


History

The singular value decomposition was originally developed by
differential geometers Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus, linear algebra and multil ...
, who wished to determine whether a real
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
could be made equal to another by independent orthogonal transformations of the two spaces it acts on. Eugenio Beltrami and
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a complete set of invariants for bilinear forms under orthogonal substitutions.
James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...
also arrived at the singular value decomposition for real square matrices in 1889, apparently independently of both Beltrami and Jordan. Sylvester called the singular values the ''canonical multipliers'' of the matrix The fourth mathematician to discover the singular value decomposition independently is Autonne in 1915, who arrived at it via the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by Carl Eckart and Gale J. Young in 1936; they saw it as a generalization of the principal axis transformation for Hermitian matrices. In 1907, Erhard Schmidt defined an analog of singular values for
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involvi ...
s (which are compact, under some weak technical assumptions); it seems he was unaware of the parallel work on singular values of finite matrices. This theory was further developed by
Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at th ...
in 1910, who is the first to call the numbers \sigma_k ''singular values'' (or in French, ''valeurs singulières''). Practical methods for computing the SVD date back to Kogbetliantz in 1954–1955 and Hestenes in 1958, resembling closely the Jacobi eigenvalue algorithm, which uses plane rotations or Givens rotations. However, these were replaced by the method of Gene Golub and William Kahan published in 1965, which uses
Householder transformation In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection (mathematics), reflection about a plane (mathematics), plane or hyperplane conta ...
s or reflections. In 1970, Golub and Christian Reinsch published a variant of the Golub/Kahan algorithm that is still the one most-used today.


See also


Notes


References

* * * * * * * * * Halldor, Bjornsson and Venegas, Silvia A. (1997)
"A manual for EOF and SVD analyses of climate data"
McGill University, CCGCR Report No. 97-1, Montréal, Québec, 52pp. * * * * * * * *


External links



{{DEFAULTSORT:Singular Value Decomposition Linear algebra Numerical linear algebra Matrix theory Matrix decompositions Functional analysis