In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ...

The spectral theorem says that normal matrices can be unitarily diagonalized using a basis of eigenvectors. The SVD can be seen as a generalization of the spectral theorem to arbitrary, not necessarily square, matrices. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... A complex square matrix A is a normal matrix if where A* is the conjugate transpose of A. (If A is a real matrix, A*=AT and so it is normal if ATA = AAT.) All unitary, hermitian, and skew-hermitian matrices are normal. ... The term diagonalization is used in two different senses in mathematics: The process of finding a diagonal matrix similar to a given square matrix or representing a given linear map. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...

Statement of the theorem

Suppose M is an m-by-n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

where U is an m-by-m unitary matrix over K, the matrix Σ is m-by-n with nonnegative numbers on the diagonal and zeros off the diagonal, and V* denotes the conjugate transpose of V, an n-by-n unitary matrix over K. Such a factorization is called a singular-value decomposition of M. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

• The matrix V thus contains a set of orthonormal "input" or "analysing" basis vector directions for M
• The matrix U contains a set of orthonormal "output" basis vector directions for M
• The matrix Σ contains the singular values, which can be thought of as scalar "gain controls" by which each corresponding input is multiplied to give a corresponding output.

A common convention is to order the values Σ_i,i in non-increasing fashion. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not). In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. ...

Certain programming languages, such as R, use a notation wherein -- for p = min(m,n) -- U is m-by-p, Σ is p-by-p, and V is n-by-p. Screenshot of Mac OS X RAqua desktop The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and graphics. ...

Example

Consider the matrix

Its singular value decomposition is

that is

Notice above that Σ only has values in its diagonal. Furthermore, as you can see below, multiplying the matrices U and V ^* by their transpose yield an identity matrix. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

and

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

A non-negative real number σ is a singular value for M if and only if there exist unit-length vectors u in K^m and v in Kⁿ such that

The vectors u and v are called left-singular and right-singular vectors for σ, respectively.

In any singular value decomposition

the diagonal entries of Σ are necessarily equal to the singular values of M. The columns of U and V are left- resp. right-singular vectors for the corresponding singular values. Consequently, the above theorem states 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 a unitary basis for K^m consisting of left-singular vectors of M.

• It is always possible to find a unitary basis for Kⁿ consisting of right-singular vectors of M.

A singular value for which we can find two left (or right) singular vectors that are not linearly dependent is called degenerate.

Non-degenerate 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 non-degenerate and non-zero, 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 non-unique singular vectors. Furthermore, if u₁ and u₂ 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. Consequently, if M has degenerate singular values, then its singular value decomposition is not unique.

Relation to eigenvalue decomposition

The singular value decomposition is very general in the sense that it can be applied to any m × n matrix. The eigenvalue decomposition, on the other hand, can only be applied to certain classes of square matrices. Nevertheless, the two decompositions are related.

In the special case that M is a Hermitian matrix which is positive semi-definite, i.e., all its eigenvalues are real and non-negative, then the singular values and singular vectors coincide with the eigenvalues and eigenvectors of M, A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

More generally, given an SVD of M, the following two relations hold:

The right hand sides of these relations describe the eigenvalue decompositions of the left hand sides. Consequently, the squares of the non-zero singular values of M are equal to the non-zero eigenvalues of either M ^* M or MM ^* . Furthermore, the columns of U (left singular vectors) are eigenvectors of MM ^* and the columns of V (right singular vectors) are eigenvectors of M ^* 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 :Rⁿ → R by f(x) = x^T M x. This continuous function attains a maximum at some u when restricted to the closed unit sphere {||x|| ≤ 1}. By the Lagrange multipliers theorem, u necessarily satisfies In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

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 real-valued 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.

Linear algebraic proof

Let M be an m-by-n matrix with complex entries. M*M is positive semidefinite, therefore Hermitian. By the spectral theorem, there exists a unitary n-by-n matrix V such that In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...

where D is diagonal and positive definite. Partition V appropriately so we can write

Therefore V*₁M*MV₁ = D, and MV₂ = 0. Define

Then

We see that this is almost the desired result, except that U₁ and V₁ are not unitary in general. U₁ is a partial isometry (U₁U*₁ = = I ) while V₁ is an isometry (V*₁V₁ = I ). To finish the argument, one simply has to "fill out" these matrices to obtain unitaries. V₂ already does this for V₁. Similarly, one can choose U₂ such that

is unitary.

Define

where the extra 0 rows are added one row for each column of U₂. 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 non-zero eigenvalues).

Variational characterization

The singular values can also be characterized as the maxima of u^TMv, 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 S^{m − 1} and S^{n − 1} denote the sets of unit 2-norm vectors in R^m and Rⁿ respectively. Define the function

for vectors u ∈ S^{m − 1} and v ∈ S^{n − 1}. Consider the function σ restricted to S^{m − 1} × S^{n − 1}. Since both S^{m − 1} and S^{n − 1} are compact sets, their product is also compact. Furthermore, since σ is continuous, it attains a largest value for at least one pair of vectors u ∈ S^{m − 1} and v ∈ S^{n − 1}. This largest value is denoted σ₁ and the corresponding vectors are denoted u₁ and v₁. Since σ₁ is the largest value of σ(u,v) it must be non-negative. If it was negative, changing the sign of either u₁ or v₁ would make it positive and thereby larger. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...

Statement: u₁, v₁ are left and right singular vectors of M with corresponding singular value σ₁.

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 σ₁ = 2 λ₁ = 2 λ₂. 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 u₁ and v₁, 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 u₁,...,u_m of U yield an orthonormal basis of K^m and the columns v₁,...,v_n of V yield an orthonormal basis of Kⁿ (with respect to the standard scalar products on these spaces). In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...

The linear transformation T :Kⁿ → 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 i-th diagonal entry of Σ, and T(v_i) = 0 for i > min(m,n). In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T :Kⁿ → K^m one can find orthonormal bases of Kⁿ and K^m such that T maps the i-th basis vector of Kⁿ to a non-negative multiple of the i-th basis vector of K^m, and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative 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 Rⁿ. The linear map T maps this sphere onto an ellipsoid in R^m. Non-zero singular values are simply the lengths of the semi-axes of this ellipsoid. Especially when n=m, and all the singular values are distinct and non-zero, the SVD decomposition 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 Rⁿ 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 Rⁿ. On a second move, apply an endomorphism d diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of T(S) as stretching coefficients. The composition d o v* then sends the unit-sphere 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.

How calculate a SVD

The SVD via the eigenvalues of M^*M

This is the algorithm that is given in the existence proof above see also http://www.uwlax.edu/faculty/will/svd/svd

Given an n by m matrix M it may be decomposed

Note U is an m by m matrix

Note V is an n by n matrix

is the diagonal matrix of eigenvalues of

Note that Sigma is an m by n matrix and the zero rows and columns make up any deficit. Explicitly D is square of dimension rank M

Note also that eigenvalues that correspond a eigenspace of more than 1 dimension are repeated: occuring once for each dimension.

is the matrix of eigenvectors of

is a matrix of orthogonal vectors spanning the kernel of

is a matrix of orthogonal vectors spanning the space

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 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<<m. The matrix U_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 decomposition of M, which can make for a significantly quicker calculation if n<<m. In linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. ...

Compact SVD

Only the r column vectors of U and r row vectors of V* corresponding to the non-zero 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<<n. The matrix U_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_r are calculated. The rest of the matrix is discarded. This can be much quicker and more economical than the thin SVD if t<<r. The matrix U_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.

Norms

Ky Fan norms

The sum of the k largest singular values of M is a matrix norm, the Ky Fan k-norm of M. In mathematics, the term Matrix Norm can have two meanings: A vector norm on matrices, i. ...

The first of the Ky Fan norms, the Ky Fan 1-norm is the same as the operator norm of M as a linear operator with respect to the Euclidean norms of K^m and Kⁿ. In other words, the Ky Fan 1-norm is the operator norm induced by the standard l² 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 M on (possibly infinite dimensional) Hilbert spaces In mathematics, the operator norm is a means to measure the size of certain linear operators. ...

But, in the matrix case, M*M^½ is a normal matrix, so ||M* M||^½ is the largest eigenvalue of M* M^½, i.e. the largest singular value of M. A complex square matrix A is a normal matrix if where A* is the conjugate transpose of A. (If A is a real matrix, A*=AT and so it is normal if ATA = AAT.) All unitary, hermitian, and skew-hermitian matrices are normal. ...

The last of the Ky Fan norms, the sum of all singular values, is the trace norm, defined by ||M|| = Tr[(M*M)^½] (the diagonal entries of M* M are the squares of the singular values). In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ...

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 <M, N> = Tr N*M. So the induced norm is ||M|| = <M, M>^½ = (Tr M*M)^½. Since trace is invariant under unitary equivalence, this shows In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H1->H2 such that there exists an orthonormal basis of H1 such that is finite. ...

where s_i are the singular values of M. This is called the Frobenius norm, Schatten 2-norm, or Hilbert-Schmidt norm of M. Direct calculation shows that if In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

the Frobenius norm of M coincides with

2DSVD

SVD computes the low-rank approximation of a single matrix (or a set of 1D vectors). This can be generalized to two-dimensional singular value decomposition (2DSVD) to do low-rank approximation of a set of 2D matrices such as a set of images or 2D weather maps. 2DSVD computes the low-rank approximation of a set of matrices such as 2D images or weather maps in a manner almost identical to SVD (singular value decomposition) which computes the low-rank approximation of a single matrix (or a set of 1D vectors). ...

HOSVD

Higher Order Singular Value Decomposition (HOSVD) is a generalization of SVD for multidimensional arrays (tensors). It is able to give optimal low-rank approximation in given dimensions of a tensor. In the case of a 3 dimensional array (array of matrices) executing HOSVD in 2 dimensions gives the same result as 2DSVD. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

It is used in multilinear signal processing and in tensor product model transformation which is the base of a modern controller design method for parameter-varying systems.

Bounded operators on Hilbert spaces

The factorization M = UΣV ^* can be extended to a bounded operator M on a separable Hilbert space H. Namely, for any bounded operator M, there exist a partial isometry U, an unitary V, a measure space (X, μ), and a non-negative measurable f such that In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In functional analysis a partial isometry is a linear map W between Hilbert spaces H, K such that there is a closed vector subspace H1 of H such that W restricted to H1 is an isometric map and W restricted to the orthogonal complement of H1 is zero. ...

where T_f is the multiplication by f on L²(X, μ). In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function &#966; is given by multiplication by a fixed function f. ...

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 calculus for self adjoint operators. The reason why U need not be unitary is because, unlike the finite dimensional case, given an isometry U₁ with non trivial kernel, a suitable U₂ may not be found such that In functional analysis, the Borel functional calculus is a functional calculus (i. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

is an unitary operator.

As for matrices, the singular value factorization is equivalent to the polar decomposition for operators: we can simply write In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ...

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/right-singular eigenvectors to the operator case, one needs to restrict to compact operators. It is a general fact that compact operators on Banach spaces, therefore Hilbert spaces, have only discrete spectrum: 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, In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are the precisely the closure of finite rank operators in the uniform operator topology. ...

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 right-singular vectors of M respectively.

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: In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are the precisely the closure of finite rank operators in the uniform operator topology. ... In functional analysis, a finite rank operator is an operator between Banach spaces whose range is finite dimensional. ...

Theorem M is compact if and only if M*M is compact.

Applications of the SVD

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 M = UΣV ^* is In mathematics, and in particular linear algebra, the pseudoinverse of an matrix is a generalization of the inverse matrix. ...

where Σ⁺ is the transpose of Σ with every nonzero entry replaced by its reciprocal. The pseudoinverse is one way to solve linear least squares problems. Linear least squares is a mathematical optimization technique to find an approximate solution for a system of linear equations that has no exact solution. ...

Range, null space and rank

Another application of the SVD is that it provides an explicit representation of the range and null space of a matrix M. The right singular vectors corresponding to vanishing singular values of M span the null space of M. The left singular vectors corresponding to the non-zero singular vectors of M span the range of M. As a consequence, the rank of M equals the number of non-zero singular values of M. Furthermore, the ranks of M, M ^* M and MM ^* must be the same. M ^* M and MM ^* have the same non-zero eigenvalues. In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, the null space (also nullspace) of an operator A is the set of all operands v which solve the equation Av = 0. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

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 non-zero singular values in a rank deficient matrix. A round-off error is the difference between the calculated approximation of a number and its exact mathematical value. ...

Matrix approximation

Some practical applications need to solve the problem of approximating a matrix M with another matrix which has a specific rank r. In the case that the approximation is based on minimizing the Frobenius norm of the difference between M and under the constraint that it turns out that the solution is given by the SVD of M, namely In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

where is the same matrix as Σ except that it contains only the r largest singular values (the other singular values are replaced by zero).

Quick proof: We hope to minimize subject to .

Suppose the SVD of M = UΣV ^* . Since the Frobenius norm is unitarily invariant, we have an equivalent statement: In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

Note that since Σ is diagonal, should be diagonal in order to minimize the Frobenius norm. Remember that the Frobenius norm is the square-root of the summation of the squared modulus of all entries. This implies that U and V are also singular matrices of . Thus we can assume that to minimize the above statement has the form: In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ... In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

where S is diagonal. The diagonal entries s_i of S are not necessarily ordered as in SVD.

From the rank constraint, i.e. S has r non-zero diagonal entries, the minimum of the above statement is obtained as follows:

Therefore, of rank r is the best approximation of M in the Frobenius norm sense when and the corresponding singular vectors are same as those of M. In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...

Other examples

The SVD is also applied extensively to the study of linear inverse problems, 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, and in signal processing and pattern recognition. It is also used in output-only modal analysis, where the non-scaled mode shapes can be determined from the singular vectors. Yet another usage is latent semantic indexing in natural language text processing. An inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained from the observed data. ... Tikhonov regularization is the most commonly used method of regularization of ill-posed problems. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In statistics, principal components analysis (PCA) is a technique that can be used to simplify a dataset; more formally it is a linear transformation that chooses a new coordinate system for the data set such that the greatest variance by any projection of the data set comes to lie on... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Pattern recognition is a field within the area of machine learning. ... Modal analysis is the study of the dynamic properties of structures under vibrational excitation. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Latent semantic analysis (LSA) is a technique in information retrieval invented in 1990 [1]. It is sometimes called latent semantic indexing (LSI). ...

One application of SVD to rather large matrices is in numerical weather prediction, 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 linearised 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. An example of 500 mb geopotential height prediction from a numerical weather prediction model Numerical weather prediction is the science of predicting the weather using mathematical models of the atmosphere. ... The Lanczos algorithm is a popular method to find a zero vector in the process of the quadratic sieve. ... Ensemble forecasting is a method used by modern operational forecast centers to account for sensitive dependency on initial conditions. ...

Computation of the SVD

The LAPACK subroutine DGESVD represents a typical approach to the computation of the singular value decomposition. If the matrix has more rows than columns a QR decomposition is first performed. The factor R is then reduced to a bidiagonal matrix. The desired singular values and vectors are then found by performing a bidiagonal QR iteration, using the LAPACK routine DBDSQR (See Demmel and Kahan for details). LAPACK, the Linear Algebra PACKage, is a software library for numerical computing written in Fortran 77. ... In linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. ... A bidiagonal matrix is a symmetric tridiagonal matrix, a special type of matrix representation from the LAPACK Fortran package. ...

The GNU Scientific Library offers three alternative ways to compute the SVD: via the Golub-Reinsch algorithm, via the modified Golub-Reinsch algorithm (faster for matrices with many more rows than columns), or via a one-sided Jacobi orthogonalization. See the GSL manual page on SVD. Code using the library and the computed results In computing, GNU Scientific Library (or GSL) is a software library written in the C programming language for numerical calculations in applied mathematics and science. ...

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 Beltrami and Camille Jordan 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 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 Eckart and Young in 1936; they saw it as a generalization of the principal axis transformation for Hermitian matrices. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... Eugenio Beltrami (16 November 1835 - 18 February 1900) was an Italian mathematician notable for his work on non-Euclidean geometry, electricity, and magnetism. ... Marie Ennemond Camille Jordan (January 5, 1838 &#8211; January 22, 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours danalyse. ... In mathematics, an invariant is something that does not change under a set of transformations. ... James Joseph Sylvester James Joseph Sylvester (September 3, 1814 London - March 15, 1897 Oxford) was an English mathematician. ... In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

In 1907, Erhard Schmidt 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 σ_k singular values (or rather, valeurs singulières). Erhard Schmidt (January 13, 1876 - December 6, 1959) was a German mathematician born in Dorpat (now Tartu, Estonia). ... In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... Charles Émile Picard (July 24, 1856 - December 11, 1941) was a leading French mathematician. ...

Practical methods for computing the SVD were unknown until 1965 when Gene Golub and William Kahan published their algorithm. In 1970, Golub and Christian Reinsch published a variant of the Golub/Kahan algorithm that is still the one most-used today. Professor Gene Howard Golub (b. ... William Velvel Kahan (born June 5, 1933, in Toronto, Ontario, Canada) is an eminent mathematician and computer scientist. ...

See also

• matrix decomposition
• s-number
• generalized singular value decomposition
• polar decomposition
• principal components analysis (PCA)
• empirical orthogonal functions (EOFs)
• canonical correlation analysis (CCA)
• latent semantic analysis

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ... In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator T acting on a Hilbert space are defined as the eigenvalues of the operator (where T* denotes the adjoint of T and the square root is taken in the operator sense). ... In linear algebra the generalized singular value decomposition (GSVD) is a matrix decomposition more general than the singular value decomposition. ... In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ... In statistics, principal components analysis (PCA) is a technique for simplifying a dataset, by reducing multidimensional datasets to lower dimensions for analysis. ... In statistics and signal processing, the method of empirical orthogonal functions is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. ... In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross-covariance matrices. ... Latent semantic analysis (LSA) is a technique in natural language processing, in particular in vectorial semantics, invented in 1990 [1] by Scott Deerwester, Susan Dumais, George Furnas, Thomas Landauer, and Richard Harshman. ...

External links

Implementations

• ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, Visual Basic, etc.
• Java SVD library routine.
• LAPACK users manual gives details of subroutines to calculate the SVD (see also [1]).
• Opencv vision library in C,C++.
• PROPACK, computes the SVD of large and sparse or structured matrices.
• SVDPACK, a library in ANSI FORTRAN 77 implementing four iterative SVD methods. Includes C and C++ interfaces.
• SVDLIBC, re-writing of SVDPACK in C.

Texts and demonstrations

• MIT Lecture series by Gilbert Strang. 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
• Java script demonstrating the SVD more extensively, paste your data from a spreadsheet.
• Chapter from "Numerical Recipes in C" gives more information about implementation and applications of SVD. (Acrobat DRM plug-in required)
• Online Matrix Calculator Performs singular value decomposition of matrices.

References

• Abdi, H. "[2] ((2007). Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD). In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
• Demmel, J. and Kahan, W. (1990). Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy. SIAM J. Sci. Statist. Comput., 11 (5), 873-912.
• Golub, G. H. and Van Loan, C. F. (1996). "Matrix Computations". 3rd ed., Johns Hopkins University Press, Baltimore. ISBN 0-8018-5414-8.
• Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, I, 211-218.
• 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.
• Hansen, P. C. (1987). The truncated SVD as a method for regularization. BIT, 27, 534-553.
• Horn, Roger A. and Johnson, Charles R (1985). "Matrix Analysis". Section 7.3. Cambridge University Press. ISBN 0-521-38632-2.
• Horn, Roger A. and Johnson, Charles R (1991). Topics in Matrix Analysis, Chapter 3. Cambridge University Press. ISBN 0-521-46713-6.
• Strang G (1998). "Introduction to Linear Algebra". Section 6.7. 3rd ed., Wellesley-Cambridge Press. ISBN 0-9614088-5-5.