In
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the
singular value decomposition (
SVD) is a
factorizationIn the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems. Example :...
of a
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
or
complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, with many useful applications in
signal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
and
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
.
Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form

where U is an m×m real or complex
unitary matrix, Σ is an m×n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and V* (the
conjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of V) is an n×n real or complex unitary matrix. The diagonal entries Σ
_{i,i} of Σ are known as the
singular values of M. The m columns of U and the n columns of V are called the
left singular vectors and
right singular vectors of M, respectively.
The singular value decomposition and the eigendecomposition are closely related. Namely:
 The left singular vectors of M are eigenvectors of
 The right singular vectors of M are eigenvectors of
 The nonzero singular values of M (found on the diagonal entries of Σ) are the square roots of the nonzero eigenvalues of or
Applications which employ the SVD include computing the pseudoinverse,
least squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
fitting of data, matrix approximation, and determining the rank, range and null space of a matrix.
Statement of the theorem
Suppose M is an m×n
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
whose entries come from the
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, which is either the field of
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s or the field of
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. Then there exists a factorization of the form

where U is an m×m
unitary matrix over K, the matrix Σ is an m×n
diagonal matrixIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
with nonnegative real numbers on the diagonal, and V*, an n×n unitary matrix over K, denotes the
conjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of V. Such a factorization is called the
singular value decomposition of M.
The diagonal entries
of Σ are known as the
singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).
Rotation, scaling, rotation
In the special but common case in which M is just an m×m square matrix with positive
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
whose entries are plain
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s, then U, V*, and Σ are m×m matrices of real numbers as well, Σ can be regarded as a scaling matrix, and U and V* can be viewed as
rotation matrices.
If the above mentioned conditions are met, the expression
can thus be intuitively interpreted as a
compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
(or
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
) of three geometrical transformations: a rotation, a
scalingIn Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
, and another rotation. For instance, the figure above explains how a
shear matrixIn mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another...
can be described as such a sequence.
Singular values as semiaxis of an ellipse or ellipsoid
As shown in the figure, the singular values can be interpreted as the semiaxes of an
ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
in 2D. This concept can be generalized to ndimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, with the singular values of any n×n square matrix being viewed as the semiaxes of an ndimensional
ellipsoid. See below for further details.
U and V are orthonormal bases
Since U and V* are unitary, the columns of each of them form a set of orthonormal vectors, which can be regarded as basis vectors. By the definition of unitary matrix, the same is true for their conjugate transposes U* and V. In short, U, U*, V, and V* are
orthonormal basesIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
.
Example
Consider the 4×5 matrix
A singular value decomposition of this matrix is given by
Notice
contains only zeros outside of the diagonal. Furthermore, because the matrices
and
are
unitary, multiplying by their respective conjugate transposes yields
identity matricesIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, as shown below. In this case, because
and
are real valued, they each are an
orthogonal matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
.
and
It should also be noted that this particular singular value decomposition is not unique. Choosing
such that
is also a valid singular value decomposition.
Singular values, singular vectors, and their relation to the SVD
A nonnegative real number σ is a
singular value for M if and only if there exist unitlength vectors u in K
^{m} and v in K
^{n} such that
The vectors u and v are called
leftsingular and
rightsingular vectors for σ, respectively.
In any singular value decomposition
the diagonal entries of Σ are equal to the singular values of M. The columns of U and V are, respectively, left and rightsingular vectors for the corresponding singular values. Consequently, the above theorem implies that:
 An m × n matrix M has at least one and at most p = min(m,n) distinct singular values.
 It is always possible to find an orthogonal basis
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal...
U for K^{m} consisting of leftsingular vectors of M.
 It is always possible to find an orthogonal basis V for K^{n} consisting of rightsingular vectors of M.
A singular value for which we can find two left (or right) singular vectors that are linearly dependent is called degenerate.
Nondegenerate singular values always have unique left and right singular vectors, up to multiplication by a unit phase factor e
^{iφ} (for the real case up to sign). Consequently, if all singular values of M are nondegenerate and nonzero, then its singular value decomposition is unique, up to multiplication of a column of U by a unit phase factor and simultaneous multiplication of the corresponding column of V by the same unit phase factor.
Degenerate singular values, by definition, have nonunique singular vectors. Furthermore, if u
_{1} and u
_{2} are two leftsingular 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. Consequently, if M has degenerate singular values, then its singular value decomposition is not unique.
Pseudoinverse
The singular value decomposition can be used for computing the pseudoinverse of a matrix. Indeed, the pseudoinverse of the matrix M with singular value decomposition
is
where Σ
^{+} is the pseudoinverse of Σ, which is formed by replacing every nonzero diagonal entry by its
reciprocalIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
and transposing the resulting matrix. The pseudoinverse is one way to solve
linear least squaresIn statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model...
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 nonzero
is to be determined which satisfies the equation. Such an
belongs to
's null space and is sometimes called a (right) null vector of
.
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 and has no vanishing singular value, the equation has no nonzero
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 nonzero
satisfying
, with
denoting the conjugate transpose of
, is called a left null vector of
.
Total least squares minimization
A total least squares problem refers to determining the vector
which minimizes the 2norm of a vector
under the constraint
. 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
rangeIn linear algebra, the column space of a matrix is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of mdimensional Euclidean space...
and
null spaceIn linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of ndimensional Euclidean space...
of a matrix M. The right singular vectors corresponding to vanishing singular values of M span the null space of M. E.g., the null space is spanned by the last two columns of
in the above example. The left singular vectors corresponding to the nonzero singular values of M span the range of M. As a consequence, the rank of M equals the number of nonzero singular values which is the same as the number of nonzero diagonal elements in
.
In numerical linear algebra the singular values can be used to determine the effective rank of a matrix, as rounding error may lead to small but nonzero singular values in a rank deficient matrix.
Lowrank matrix approximation
Some practical applications need to solve the problem of approximating a matrix
with another matrix
which has a specific rank
. In the case that the approximation is based on minimizing the Frobenius norm of the difference between
and
under the constraint that
it turns out that the solution is given by the SVD of
, namely
where
is the same matrix as
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 ).
Quick proof: We hope to minimize
subject to
.
Suppose the SVD of
. Since the Frobenius norm is unitarily invariant, we have an equivalent statement:
Note that since
is diagonal,
should be diagonal in order to minimize the Frobenius norm. Remember that the Frobenius norm is the squareroot of the summation of the squared modulus of all entries.
This implies that
and
are also singular matrices of
. Thus we can assume that
to minimize the above statement has the form:
where
is diagonal. The diagonal entries
of
are not necessarily ordered as in SVD.
From the rank constraint, i.e.
has
nonzero diagonal entries, the minimum of the above statement is obtained as follows:
Therefore,
of rank
is the best approximation of
in the Frobenius norm sense when
and the corresponding singular vectors are same as those of
.
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 productIn linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
of two vectors
, or, in coordinates,
. Specifically, the matrix M can be decomposed as:

Here
and
are the i
^{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 nonzero
is exactly the rank of the matrix.
Separable models often arise in biological systems, and the SVD decomposition is useful to analyze such systems. For example, some visual area V1 simple cells receptive fields can be well described by a
Gabor filterIn image processing, a Gabor filter, named after Dennis Gabor, is a linear filter used for edge detection. Frequency and orientation representations of Gabor filters are similar to those of the human visual system, and they have been found to be particularly appropriate for texture representation...
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 U in the SVD decomposition is then a Gabor while the first column of V represents the time modulation (or viceversa). One may then define an index of separability,
, 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
to determine the
orthogonal matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
closest to
. The closeness of fit is measured by the Frobenius norm 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.
A similar problem, with interesting applications in
shape analysisThis article describes shape analysis to analyze and process geometric shapes.The shape analysis described here is related to the statistical analysis of geometric shapes, to shape matching and shape recognition...
, is the
orthogonal Procrustes problemThe orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A and B and asked to find an orthogonal matrix R which most closely maps A to B...
, which consists of finding an orthogonal matrix
which most closely maps
to
. Specifically,
where
denotes the Frobenius norm.
This problem is equivalent to finding the nearest orthogonal matrix to a given matrix
.
The Kabsch Algorithm
The
Kabsch algorithmThe Kabsch algorithm, named after Wolfgang Kabsch, is a method for calculating the optimal rotation matrix that minimizes the RMSD between two paired sets of points...
(called
Wahba's problemIn applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix between two coordinate systems from a set of vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers...
in other fields) uses SVD to compute the optimal rotation (with respect to leastsquares 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.
Other examples
The SVD is also applied extensively to the study of linear
inverse problemAn inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
s, and is useful in the analysis of regularization methods such as that of
TikhonovTikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of illposed problems. In statistics, the method is known as ridge regression, and, with multiple independent discoveries, it is also variously known as the TikhonovMiller method, the...
. It is widely used in
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
where it is related to principal component analysis and to
Correspondence analysisCorrespondence analysis is a multivariate statistical technique proposed by Hirschfeld and later developed by JeanPaul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data...
, and in
signal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
and
pattern recognitionIn machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...
. It is also used in outputonly
modal analysisModal analysis is the study of the dynamic properties of structures under vibrational excitation.Modal analysis is the field of measuring and analysing the dynamic response of structures and or fluids when excited by an input...
, where the nonscaled mode shapes can be determined from the singular vectors. Yet another usage is
latent semantic indexingLatent Semantic Indexing is an indexing and retrieval method that uses a mathematical technique called Singular value decomposition 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...
in natural language text processing.
The SVD also plays a crucial role in the field of
Quantum informationIn quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a twolevel quantum system...
, in a form often referred to as the
Schmidt decompositionIn linear algebra, the Schmidt decomposition refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has applications in quantum information theory and plasticity....
. 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
matrix is larger than one.
One application of SVD to rather large matrices is in
numerical weather predictionNumerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...
, where
Lanczos methodThe Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very...
s 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 forecastEnsemble forecasting is a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of a dynamical system...
, giving a handle on some of the uncertainty that should be allowed for around the current central prediction.
Another application of SVD for daily life is that point in perspective view can be unprojected in a photo using the calculated SVD matrix, this application leads to measuring length (a.k.a. the distance of two unprojected points in perspective photo) by marking out the 4 corner points of knownsize object in a single photo. PRuler is a demo to implement this application by taking a photo of a regular credit card
Relation to eigenvalue decomposition
The singular value decomposition is very general in the sense that it can be applied to any m × n matrix whereas eigenvalue decomposition can only be applied to certain classes of square matrices. Nevertheless, the two decompositions are related.
Given an SVD of M, as described above, the following two relations hold:
The right hand sides of these relations describe the eigenvalue decompositions of the left hand sides. Consequently:
 The columns of V (right singular vectors) are eigenvectors of
 The columns of U (left singular vectors) are eigenvectors of
 The nonzero elements of Σ (nonzero singular values) are the square roots of the nonzero eigenvalues of or
In the special case that M is a
normal matrixA complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...
, which by definition must be square, the spectral theorem says that it can be unitarily
diagonalizedIn linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
using a basis of eigenvectors, so that it can be written
for a unitary matrix U and a diagonal matrix D. When M is also
positive semidefiniteIn linear algebra, a positivedefinite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positivedefinite symmetric bilinear form ....
, the decomposition
is also a singular value decomposition.
However, the eigenvalue decomposition and the singular value decomposition differ for all other matrices M: the eigenvalue decomposition is
where U is not necessarily unitary and D is not necessarily positive semidefinite, while the SVD is
where Σ is a diagonal positive semidefinite, and U and V are unitary matrices that are not necessarily related except through the matrix M.
Existence
An eigenvalue λ of a matrix is characterized by the algebraic relation M u = λ u. When M is
Hermitian, a variational characterization is also available. Let M be a real n × n
symmetric matrix. Define f :
R^{n} →
R by f(x) = x
^{T} M x. By the
extreme value theoremIn calculus, the extreme value theorem states that if a realvalued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
, this continuous function attains a maximum at some u when restricted to the closed unit sphere {x ≤ 1}. By the
Lagrange multipliersIn mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
theorem, u necessarily satisfies
where the nabla symbol,
, is the
delIn vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a onedimensional domain, it denotes its standard derivative as defined in calculus...
operator.
A short calculation shows the above leads to M u = λ u (symmetry of M is needed here). Therefore λ is the largest eigenvalue of M. The same calculation performed on the orthogonal complement of u gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there f(x) = x* M x is a realvalued function of 2n 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 M is no longer required.
This section gives these two arguments for existence of singular value decomposition.
Based on the spectral theorem
Let M be an mbyn matrix with complex entries. M*M is positive semidefinite and Hermitian. By the
spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
, there exists a unitary nbyn matrix V such that
where D is diagonal and positive definite. Partition V appropriately so we can write
Therefore V
_{1}*M*MV
_{1} = D and V
_{2}*M*MV
_{2} = 0. The latter means MV
_{2} = 0.
Also, since V is unitary, V
_{1}*V
_{1} = I, V
_{2}*V
_{2} = I and V
_{1}V
_{1}* + V
_{2}V
_{2}* = I.
Define
Then
We see that this is almost the desired result, except that U
_{1} and V
_{1} are not unitary in general, but merely
isometriesIn mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
. To finish the argument, one simply has to "fill out" these matrices to obtain unitaries. For example, one can choose U
_{2} such that
is unitary.
Define
where extra zero rows are added
or removed to make the number of zero rows equal the number of columns of U
_{2}. Then
which is the desired result:
Notice the argument could begin with diagonalizing MM* rather than M*M (This shows directly that MM* and M*M have the same nonzero eigenvalues).
Based on variational characterization
The singular values can also be characterized as the maxima of u
^{T}Mv, considered as a function of u and v, over particular subspaces. The singular vectors are the values of u and v where these maxima are attained.
Let M denote an m × n matrix with real entries. Let
and
denote the sets of unit 2norm vectors in
R^{m} and
R^{n} respectively. Define the function
for vectors u ∈
and v ∈
. Consider the function σ restricted to
×
. Since both
and
are
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
sets, their
productIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
is also compact. Furthermore, since σ is continuous, it attains a largest value for at least one pair of vectors u ∈
and v ∈
. This largest value is denoted σ
_{1} and the corresponding vectors are denoted u
_{1} and v
_{1}. Since
is the largest value of
it must be nonnegative. If it were negative, changing the sign of either u
_{1} or v
_{1} would make it positive and therefore larger.
Statement: u
_{1}, v
_{1} are left and right singular vectors of M with corresponding singular value σ
_{1}.
Proof: Similar to the eigenvalues case, by assumption the two vectors satisfy the Lagrange multiplier equation:
After some algebra, this becomes
and
Multiplying the first equation from left by
and the second equation from left by
and taking u = v = 1 into account gives
So σ
_{1} = 2 λ
_{1} = 2 λ
_{2}. By properties of the functional φ defined by
we have
Similarly,
This proves the statement.
More singular vectors and singular values can be found by maximizing σ(u, v) over normalized u, v which are orthogonal to u
_{1} and v
_{1}, respectively.
The passage from real to complex is similar to the eigenvalue case.
Geometric meaning
Because U and V are unitary, we know that the columns u
_{1},...,u
_{m} of U yield an
orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of K
^{m} and the columns v
_{1},...,v
_{n}
of V yield an orthonormal basis of K
^{n} (with respect to the standard scalar products on these spaces).
The
linear transformationIn mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
T :K
^{n} → K
^{m} that takes a vector x to Mx has a particularly simple description with respect to these orthonormal bases: we have T(v
_{i}) = σ
_{i} u
_{i}, for i = 1,...,min(m,n), where σ
_{i} is the ith diagonal entry of Σ, and T(v
_{i}) = 0 for i > min(m,n).
The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T :K
^{n} → K
^{m} one can find orthonormal bases of K
^{n} and K
^{m} such that T maps the ith basis vector of K
^{n} to a nonnegative multiple of the ith basis vector of K
^{m}, and sends the leftover basis vectors to zero.
With respect to these bases, the map T is therefore represented by a diagonal matrix with nonnegative real diagonal entries.
To get a more visual flavour of singular values and SVD decomposition —at least when working on real vector spaces— consider the sphere S of radius one in
R^{n}. The linear map T maps this sphere onto an
ellipsoid in
R^{m}. Nonzero singular values are simply the lengths of the
semiaxesIn geometry, the semiminor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semimajor axis...
of this ellipsoid. Especially when n=m, and all the singular values are distinct and nonzero, the SVD of the linear map T can be easily analysed as a succession of three consecutive moves : consider the ellipsoid T(S) and specifically its axes ; then consider the directions in
R^{n} sent by T onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry v* sending these directions to the coordinate axes of
R^{n}. On a second move, apply an
endomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
d diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semiaxes lengths of T(S) as stretching coefficients. The composition d
o v* then sends the unitsphere onto an ellipsoid isometric to T(S). To define the third and last move u, just apply an isometry to this ellipsoid so as to carry it over T(S). As can be easily checked, the composition u
o d
o v* coincides with T.
Numerical Approach
The SVD of a matrix M is typically computed by a twostep procedure. In the first step, the matrix is reduced to a
bidiagonal matrixA bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal above or the diagonal below.So that means there are two non zero diagonal in the matrix....
. This takes O(mn
^{2}) floatingpoint operations, assuming that m ≥ n (this formulation uses the
big O notationIn mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, BachmannLandau notation, or...
). The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an
iterative methodIn computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
(as with
eigenvalue algorithmIn linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.Characteristic polynomial:...
s). However, in practice it suffices to compute the SVD up to a certain precision, like the
machine epsilonMachine epsilon gives an upper bound on the relative error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of computational science...
. If this precision is considered constant, then the second step takes O(n) iterations, each costing O(n) flops. Thus, the first step is more expensive, and the overall cost is O(mn
^{2}) flops .
The first step can be done using Householder reflections for a cost of 4mn
^{2} − 4n
^{3}/3 flops, assuming that only the singular values are needed and not the singular vectors. If m is much larger than n then it is advantageous to first reduce the matrix M to a triangular matrix with the
QR decompositionIn linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
and then use Householder reflections to further reduce the matrix to bidiagonal form; the combined cost is 2mn
^{2} + 2n
^{3} flops .
The second step can be done by a variant of the
QR algorithmIn numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR transformation was developed in the late 1950s by John G.F. Francis and by Vera N. Kublanovskaya , working independently...
for the computation of eigenvalues, which was first described by . The
LAPACKExternal links:* : a modern replacement for PLAPACK and ScaLAPACK* on Netlib.org* * * : a modern replacement for LAPACK that is MultiGPU ready* on Sourceforge.net* * optimized LAPACK for Solaris OS on SPARC/x86/x64 and Linux* * *...
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 LibraryIn computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...
(GSL). The GSL also offers an alternative method, which uses a onesided Jacobi orthogonalization in step 2 . This method computes the SVD of the bidiagonal matrix by solving a sequence of 2by2 SVD problems, similar to how the
Jacobi eigenvalue algorithmIn numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
solves a sequence of 2by2 eigenvalue methods . Yet another method for step 2 uses the idea of
divideandconquer eigenvalue algorithmDivideandconquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is...
s .
Analytic Result of 2by2 SVD
The singular values of a 2by2 matrix can be found analytically. Let the matrix be
where
are complex numbers that parameterize the matrix, and
denote the
Pauli matricesThe Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
. Then its two singular values are given by
Reduced SVDs
In applications it is quite unusual for the full SVD, including a full unitary decomposition of the nullspace 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 m×n matrix M of rank r:
Thin SVD
Only the n column vectors of U corresponding to the row vectors of V* are calculated. The remaining column vectors of U are not calculated. This is significantly quicker and more economical than the full SVD if n<
n is thus m×n, Σ_{n} is n×n diagonal, and V is n×n.
The first stage in the calculation of a thin SVD will usually be a QR decompositionIn linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
of M, which can make for a significantly quicker calculation if n<
Compact SVD
Only the r column vectors of U and r row vectors of V* corresponding to the nonzero singular values Σ_{r} are calculated. The remaining vectors of U and V* are not calculated. This is quicker and more economical than the thin SVD if r<r is thus m×r, Σ_{r} is r×r diagonal, and V_{r}* is r×n.
Truncated SVD
Only the t column vectors of U and t row vectors of V* corresponding to the t largest singular values Σ_{t} are calculated. The rest of the matrix is discarded. This can be much quicker and more economical than the compact SVD if t<t is thus m×t, Σ_{t} is t×t diagonal, and V_{t}* is t×n'.
Of course the truncated SVD is no longer an exact decomposition of the original matrix M, but as discussed below, the approximate matrix is in a very useful sense the closest approximation to M that can be achieved by a matrix of rank t.
Ky Fan norms
The sum of the k largest singular values of M is a matrix norm, the Ky FanKy Fan was an American mathematician and Emeritus Professor of Mathematics at the University of California, Santa Barbara .Biography:...
knorm of M.
The first of the Ky Fan norms, the Ky Fan 1norm is the same as the operator normIn mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition :...
of M as a linear operator with respect to the Euclidean norms of K^{m} and K^{n}. In other words, the Ky Fan 1norm is the operator norm induced by the standard l^{2} Euclidean inner product. For this reason, it is also called the operator 2norm. One can easily verify the relationship between the Ky Fan 1norm and singular values. It is true in general, for a bounded operator M on (possibly infinite dimensional) Hilbert spaces
But, in the matrix case, M*M^{½} is a normal matrixA complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...
, so M* M^{½} is the largest eigenvalue of M* M^{½}, i.e. the largest singular value of M.
The last of the Ky Fan norms, the sum of all singular values, is the trace normIn mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
(also known as the 'nuclear norm'), defined by M = Tr[(M*M)^{½}] (the diagonal entries of M* M 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 n × n matrices, defined by . So the induced norm is . Since trace is invariant under unitary equivalence, this shows
where are the singular values of M. This is called the Frobenius norm, Schatten 2norm, or Hilbert–Schmidt norm of M. Direct calculation shows that if
the Frobenius norm of M coincides with
Tensor SVD
Unfortunately, the problem of finding a low rank approximation to a tensor is illposed. In other words, there doesn't exist a best possible solution, but instead a sequence of better and better approximations that converge to infinitely large matrices. But in spite of this, there are several ways of attempting this decomposition.
There exist two types of tensor decompositions which generalise SVD to multiway arrays. One decomposition decomposes a tensor into a sum of rank1 tensors, see CandecompPARAFAC (CP) algorithm. The CP algorithm should not be confused with a rankR decomposition but, for a given N, it decomposes a tensor into a sum of N rank1 tensors that optimally fit the original tensor. The second type of decomposition computes the orthonormal subspaces associated with the different axes or modes of a tensor (orthonormal row space, column space, fiber space, etc.). This decomposition is referred to in the literature as the Tucker3/TuckerMIn mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tuckeralthough it goes back to Hitchcock in 1927....
, Mmode SVD, multilinear SVD and sometimes referred to as a higherorder SVD (HOSVD). In addition, multilinear principal component analysisMultilinear principalcomponent analysis is a mathematical procedure that uses multiple orthogonal transformations to convert a set of multidimensional objects into another set of multidimensional objects of lower dimensions. There is one orthogonal transformation for each dimension...
in multilinear subspace learningMultilinear subspace learning aims to learn a specific small part of a large space of multidimensional objects having a particular desired property. It is a dimensionality reduction approach for finding a lowdimensional representation with certain preferred characteristics of highdimensional...
involves the same mathematical operations as Tucker decomposition, being used in a different context of dimensionality reductionIn machine learning, dimension reduction is the process of reducing the number of random variables under consideration, and can be divided into feature selection and feature extraction.Feature selection:...
.
Bounded operators on Hilbert spaces
The factorization can be extended to a bounded operatorIn functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all nonzero vectors v in X...
M on a separable Hilbert space H. Namely, for any bounded operator M, there exist a partial isometryIn functional analysis a partial isometry is a linear map W between Hilbert spaces H and K such that the restriction of W to the orthogonal complement of its kernel is an isometry...
U, a unitary V, a measure space (X, μ), and a nonnegative measurable f such that
where is the multiplication by fIn operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...
on L^{2}(X, μ).
This can be shown by mimicking the linear algebraic argument for the matricial case above. VT_{f} V* is the unique positive square root of M*M, as given by the Borel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
for self adjoint operators. The reason why U need not be unitary is because, unlike the finite dimensional case, given an isometry U_{1} with non trivial kernel, a suitable U_{2} may not be found such that
is a unitary operator.
As for matrices, the singular value factorization is equivalent to the polar decomposition for operators: we can simply write
and notice that U V* is still a partial isometry while VT_{f} V* is positive.
Singular values and compact operators
To extend notion of singular values and left/rightsingular vectors to the operator case, one needs to restrict to compact operatorsIn functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finiterank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
. It is a general fact that compact operators on Banach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm · such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s have only discrete spectrum. This is also true for compact operators on Hilbert spaces, since Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
s are a special case of Banach spaces. If T is compact, every nonzero λ in its spectrum is an eigenvalue. Furthermore, a compact self adjoint operator can be diagonalized by its eigenvectors. If M is compact, so is M*M. Applying the diagonalization result, the unitary image of its positive square root T_{f} has a set of orthonormal eigenvectors {e_{i}} corresponding to strictly positive eigenvalues {σ_{i}}. For any ψ ∈ H,
where the series converges in the norm topology on H. Notice how this resembles the expression from the finite dimensional case. The σ_{i} 's are called the singular values of M. {U e_{i}} and {V e_{i}} can be considered the left and rightsingular vectors of M respectively.
Compact operators on a Hilbert spaceIn functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finiterank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
are the closure of finiterank operators in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:
Theorem M is compact if and only if M*M is compact.
History
The singular value decomposition was originally developed by differential geometers, who wished to determine whether a real bilinear form could be made equal to another by independent orthogonal transformations of the two spaces it acts on. Eugenio BeltramiEugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...
and Camille JordanMarie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a complete setIn mathematics, a complete set of invariants for a classification problem is a collection of mapsf_i : X \to Y_i \,, such that x ∼ x' if and only if f_i = f_i for all i...
of invariantIn mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
s for bilinear forms under orthogonal substitutions. James Joseph SylvesterJames Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
also arrived at the singular value decomposition for real square matrices in 1889, apparently independent of both Beltrami and Jordan. Sylvester called the singular values the canonical multipliers of the matrix A. The fourth mathematician to discover the singular value decomposition independently is Autonne in 1915, who arrived at it via the polar decomposition. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by Carl EckartCarl Henry Eckart was an American physicist, physical oceanographer, geophysicist, and administrator. He codeveloped the WignerEckart theorem and is also known for the Eckart conditions in quantum mechanics.Education:Eckart began college in 1919 at Washington University in St...
and Gale Young in 1936; they saw it as a generalization of the principal axis transformation for Hermitian matrices.
In 1907, Erhard SchmidtErhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from GeorgAugust University of Göttingen in 1905...
defined an analog of singular values for integral operators (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 in 1910, who is the first to call the numbers singular values (or rather, valeurs singulières).
Practical methods for computing the SVD date back to KogbetliantzErvand George Kogbetliantz Dr. Kogbetliantz was an Armenian/American mathematician and the first president of the Yerevan State University. He left Russia in 1918. He received a Doctorate in mathematics from the University of Paris in 1923...
in 1954, 1955 and HestenesMagnus Rudolph Hestenes was an American mathematician. Together with Cornelius Lanczos and Eduard Stiefel, he invented the conjugate gradient method....
in 1958. resembling closely the Jacobi eigenvalue algorithmIn numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
, which uses plane rotations or Givens rotations. However, these were replaced by the method of Gene GolubGene Howard Golub , Fletcher Jones Professor of Computer Science at Stanford University, was one of the preeminent numerical analysts of his generation....
and William KahanWilliam Morton Kahan is a mathematician and computer scientist who received the Turing Award in 1989 for "his fundamental contributions to numerical analysis", and was named an ACM Fellow in 1994....
published in 1965, which uses Householder transformationIn linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm...
s or reflections.
In 1970, Golub and Christian Reinsch published a variant of the Golub/Kahan algorithm that is still the one mostused today.
See also
 Canonical correlation analysis (CCA)
 Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
 Correspondence analysis
Correspondence analysis is a multivariate statistical technique proposed by Hirschfeld and later developed by JeanPaul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data...
(CA)
 Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing highdimensional spaces that do not occur in lowdimensional settings such as the physical space commonly modeled with just three dimensions.There are multiple phenomena referred to by this name in...
 Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
 Dimension reduction
 Eigendecomposition
 Empirical orthogonal functions
In statistics and signal processing, the method of empirical orthogonal function analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. It is the same as performing a principal components analysis on the data, except that the...
(EOFs)
 Fourier analysis
 Fourierrelated transforms
 Generalized singular value decomposition
In linear algebra the generalized singular value decomposition is a matrix decomposition more general than the singular value decomposition...
 Latent semantic analysis
Latent semantic analysis is a technique in natural language processing, in particular in vectorial semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the documents and terms. LSA assumes that words that are close...
 Latent semantic indexing
Latent Semantic Indexing is an indexing and retrieval method that uses a mathematical technique called Singular value decomposition 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...
 Linear least squares
In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model...
 Locality sensitive hashing
Localitysensitive hashing is a method of performing probabilistic dimension reduction of highdimensional data. The basic idea is to hash the input items so that similar items are mapped to the same buckets with high probability .Definition:An LSH family \mathcal F is defined fora...
 Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems. Example :...
 Multilinear principal component analysis
Multilinear principalcomponent analysis is a mathematical procedure that uses multiple orthogonal transformations to convert a set of multidimensional objects into another set of multidimensional objects of lower dimensions. There is one orthogonal transformation for each dimension...
(MPCA)
 Nearest neighbor search
Nearest neighbor search , also known as proximity search, similarity search or closest point search, is an optimization problem for finding closest points in metric spaces. The problem is: given a set S of points in a metric space M and a query point q ∈ M, find the closest point in S to q...
 Nonlinear iterative partial least squares
In statistics, nonlinear iterative partial least squares is an algorithm for computing the first few components in a principal component or partial least squares analysis. For very highdimensional datasets, such as those generated in the 'omics sciences it is usually only necessary to compute...
 Polar decomposition
 Principal components analysis
Principal component analysis is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to...
(PCA)
 Singular value
 Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
 von Neumann's trace inequality
 Wavelet compression
Libraries that support complex and real SVD
 LAPACK
External links:* : a modern replacement for PLAPACK and ScaLAPACK* on Netlib.org* * * : a modern replacement for LAPACK that is MultiGPU ready* on Sourceforge.net* * optimized LAPACK for Solaris OS on SPARC/x86/x64 and Linux* * *...
(website), the Linear Algebra Package. The user manual gives details of subroutines to calculate the SVD (see also http://www.netlib.org/lapack/lug/node32.html).
 LINPACK Z (website), Linear Algebra Library. Has officially been superseded by LAPACK, but it includes a C version of SVD for complex numbers.
 For the Python
Python is a generalpurpose, highlevel programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
programming language:
 NumPy (NumPy is module for numerical computing with arrays and matrices)
 SciPy (SciPy
SciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...
contains many numerical routines)
Libraries that support real SVD
 GNU Scientific Library
In computing, the GNU Scientific Library is a software library written in the C programming language for numerical calculations in applied mathematics and science...
(website), a numerical C/C++ library supporting SVD (see http://www.gnu.org/software/gsl/manual/html_node/SingularValueDecomposition.html).
 For the Python
Python is a generalpurpose, highlevel programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
programming language:
 NumPy (NumPy is module for numerical computing with arrays and matrices)
 SciPy (SciPy
SciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...
contains many numerical routines)
 Gensim, efficient randomized algorithm on top of NumPy; unlike other implementations, allows SVD of matrices larger than RAM (incremental online SVD).
 sparsesvd, Python wrapper of SVDLIBC.
 SVDPython, pure Python SVD under GNU GPL.
 ALGLIB, includes a partial port of the LAPACK to C++, C#, Delphi, Visual Basic, etc.
 JAMA, a Java matrix package provided by the NIST.
 COLT, a Java package for High Performance Scientific and Technical Computing, provided by CERN
The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
.
 Eigen, a templated C++ implementation.
 redsvd, efficient randomized algorithm on top of C++ Eigen.
 PROPACK, computes the SVD of large and sparse or structured matrices, in Fortran 77.
 SVDPACK, a library in ANSI FORTRAN 77 implementing four iterative SVD methods. Includes C and C++ interfaces.
 SVDLIBC, rewriting of SVDPACK in C, with minor bug fixes.
 SVDLIBJ, a Java port of SVDLIBC. (Also available as an executable .jar similar to SVDLIBC in the SSpace Package)
 SVDLIBC# SVDLIBC converted to C#.
 dANN part of the linear algebra package of the dANN
Dann may refer to:* Dann, the surname* dANN, the artificial intelligence software libraryDann as a surname may refer to:* Mary Dann and Carrie Dann, Native American activists* Colin Dann, a British author* Jack Dann, a American writer...
java Artificial IntelligenceArtificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
library by Syncleus, Inc.
 GraphLab GraphLab collaborative filtering library, large scale parallel implementation of SVD (in C++) for multicore.
Texts and demonstrations
 MIT Lecture series by Gilbert Strang
William Gilbert Strang , usually known as simply Gilbert Strang or Gil Strang, is a renowned American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra...
. See Lecture #29 on the SVD (scroll down to the bottom till you see "Singular Value Decomposition"). The first 17 minutes give the overview. Then Prof. Strang works two examples. Then the last 4 minutes (min 36 to min 40) are a summary. You can probably fast forward the examples, but the first and last are an excellent concise visual presentation of the topic.
 Applications of SVD on PC Hansen's web site.
 Introduction to the Singular Value Decomposition by Todd Will of the University of Wisconsin—La Crosse. This site has animations for the visual minded as well as demonstrations of compression using SVD.
 Los Alamos group's book chapter has helpful gene data analysis examples.
 SVD, another explanation of singular value decomposition
 SVD Tutorial, yet another explanation of SVD. Very intuitive.
 Javascript script demonstrating SVD more extensively, paste your data from a spreadsheet.
 http://www.stasegem.be/shop2/SVD.htm demonstrating SVD recommender system (same as above but how to make your own recommender matrix
 Chapter from "Numerical Recipes in C" gives more information about implementation and applications of SVD. (Acrobat DRM plugin required)
 Online Matrix Calculator Performs singular value decomposition of matrices.
 A simple tutorial on SVD and applications of Spectral Methods
 Matrix and Tensor Decompositions in Genomic Signal Processing
 SVD on MathWorld, with image compression as an example application.
 Notes on RankK Approximation (and SVD for the uninitiated) at The University of Texas at Austin. This demo with Octave uses the data file lenna.m.
 If you liked this... New York Times article on SVD in movieratings and Netflix
Netflix, Inc., is an American provider of ondemand internet streaming media in the United States, Canada, and Latin America and flat rate DVDbymail in the United States. The company was established in 1997 and is headquartered in Los Gatos, California...
 David Austin, We Recommend a Singular Value Decomposition, Featured Column from the AMS, August 2009.
Songs
 It Had To Be U is a song, written by Michael Greenacre, about the singular value decomposition, explaining its definition and role in statistical dimension reduction. It was first performed at the joint meetings of the 9th Tartu Conference on Multivariate Statistics and 20th International Workshop on Matrices and Statistics, in Tartu, Estonia, June 2011.