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In linear algebra

Linear algebra

Linear 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 factorization

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 :...

of a real

Real number

In 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 complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

matrix

Matrix (mathematics)

In 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 processing

Signal processing

Signal 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 statistics

Statistics

Statistics 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 transpose

Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, 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...

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 non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of or

Applications which employ the SVD include computing the pseudoinverse, least squares

Least squares

The 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 matrix

Matrix (mathematics)

whose entries come from the field

Field (mathematics)

In 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 number

Real number

In 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 number

Complex number

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 matrix

Diagonal matrix

In 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 transpose

Conjugate transpose

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 determinant

Determinant

In 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 number

Real number

In 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 composition

Function composition

In 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 sequence

Sequence

In 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 scaling

Scaling (geometry)

In 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 matrix

Shear matrix

In 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 ellipse

Ellipse

In 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 2-D. This concept can be generalized to n-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional 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 n-dimensional 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 bases

Orthonormal basis

In 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 matrices

Identity matrix

In 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 matrix

Orthogonal matrix

In 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 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 equal to the singular values of M. The columns of U and V are, respectively, left- and right-singular 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
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 left-singular vectors of M.

It is always possible to find an orthogonal basis V 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 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.

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

where Σ⁺ is the pseudoinverse of Σ, which is formed by replacing every nonzero diagonal entry by its reciprocal

Multiplicative inverse

In 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 squares

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...

problems.

Solving homogeneous linear equations

A set of homogeneous linear equations can be written as

for a matrix

and vector

. A typical situation is that

is known and a non-zero

is to be determined which satisfies the equation. Such an

belongs to

's null space 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 non-zero

as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right singular vectors is a valid solution. Analogously
to the definition of a (right) null vector, a non-zero

satisfying

, with

denoting the conjugate transpose of

, is called a left null vector of

Total least squares minimization

A total least squares problem refers to determining the vector

which minimizes the 2-norm 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 range

Column space

In 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 m-dimensional Euclidean space...

and null space

Null space

In 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 n-dimensional 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 non-zero singular values of M span the range of M. As a consequence, the rank of M equals the number of non-zero singular values which is the same as the number of non-zero 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 non-zero singular values in a rank deficient matrix.

Low-rank 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 square-root 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

are not necessarily ordered as in SVD.

From the rank constraint, i.e.

has

non-zero 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 product

Outer product

In 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 non-zero

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 filter

Gabor filter

In 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 vice-versa). 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 matrix

Orthogonal matrix

In 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 analysis

Shape analysis

This 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 problem

Orthogonal Procrustes problem

The 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

. 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 algorithm

Kabsch algorithm

The 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 problem

Wahba's problem

In 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 least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.

Other examples

The SVD is also applied extensively to the study of linear inverse problem

Inverse problem

An 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 Tikhonov

Tikhonov regularization

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems. In statistics, the method is known as ridge regression, and, with multiple independent discoveries, it is also variously known as the Tikhonov-Miller method, the...

. It is widely used in statistics

Statistics

where it is related to principal component analysis and to Correspondence analysis

Correspondence analysis

Correspondence analysis is a multivariate statistical technique proposed by Hirschfeld and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data...

, and in signal processing

Signal processing

Signal 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 recognition

Pattern recognition

In 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 output-only modal analysis

Modal analysis

Modal 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 non-scaled mode shapes can be determined from the singular vectors. Yet another usage is latent semantic indexing

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...

in natural language text processing.

The SVD also plays a crucial role in the field of Quantum information

Quantum information

In 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 two-level quantum system...

, in a form often referred to as the Schmidt decomposition

Schmidt decomposition

In 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 prediction

Numerical weather prediction

Numerical 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 method

Lanczos algorithm

The 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 forecast

Ensemble forecasting

Ensemble 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 known-size 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 non-zero elements of Σ (non-zero singular values) are the square roots of the non-zero eigenvalues of or

In the special case that M is a normal matrix

Normal matrix

A 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 diagonalized

Diagonalizable matrix

In 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 semi-definite

Positive-definite matrix

In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite 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 semi-definite, while the SVD is

where Σ is a diagonal positive semi-definite, 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 :Rⁿ → R by f(x) = x^T M x. By the extreme value theorem

Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued 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 multipliers

Lagrange multipliers

In 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 del

Del

In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional 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 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.

Based on the spectral theorem

Let M be an m-by-n matrix with complex entries. M*M is positive semidefinite and Hermitian. By the spectral theorem

Spectral theorem

In 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 n-by-n matrix V such that

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

Therefore V₁*M*MV₁ = D and V₂*M*MV₂ = 0. The latter means MV₂ = 0.

Also, since V is unitary, V₁*V₁ = I, V₂*V₂ = I and V₁V₁* + V₂V₂* = I.

Define

Then

We see that this is almost the desired result, except that U₁ and V₁ are not unitary in general, but merely isometries

Isometry

In mathematics, an isometry is a distance-preserving 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 U₂ 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 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).

Based on 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

and

denote the sets of unit 2-norm vectors in R^m and Rⁿ respectively. Define the function

for vectors u ∈

and v ∈

. Consider the function σ restricted to

. Since both

and

are compact

Compact space

In 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 product

Product topology

In 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 σ₁ and the corresponding vectors are denoted u₁ and v₁. Since

is the largest value of

it must be non-negative. If it were negative, changing the sign of either u₁ or v₁ would make it positive and therefore larger.

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

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).

The linear transformation

Linear transformation

In 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 :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).

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

Semi-minor axis

In geometry, the semi-minor 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 semi-major axis...

of this ellipsoid. Especially when n=m, and all the singular values are distinct and non-zero, 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 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

Endomorphism

In 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 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.

Numerical Approach

The SVD of a matrix M is typically computed by a two-step procedure. In the first step, the matrix is reduced to a bidiagonal matrix

Bidiagonal matrix

A bidiagonal matrix is a matrix with non-zero 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(mn²) floating-point operations, assuming that m ≥ n (this formulation uses the big O notation

Big O notation

In 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, Bachmann-Landau notation, or...

). The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an iterative method

Iterative method

In 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 algorithm

Eigenvalue algorithm

In 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 epsilon

Machine epsilon

Machine 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(mn²) flops .

The first step can be done using Householder reflections for a cost of 4mn² − 4n³/3 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 decomposition

QR decomposition

In 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 2mn² + 2n³ flops .

The second step can be done by a variant of the QR algorithm

QR algorithm

In 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 LAPACK

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* * *...

subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small . Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition.

The same algorithm is implemented in the GNU Scientific Library

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...

(GSL). The GSL also offers an alternative method, which uses a one-sided Jacobi orthogonalization in step 2 . This method computes the SVD of the bidiagonal matrix by solving a sequence of 2-by-2 SVD problems, similar to how the Jacobi eigenvalue algorithm

Jacobi eigenvalue algorithm

In 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 2-by-2 eigenvalue methods . Yet another method for step 2 uses the idea of divide-and-conquer eigenvalue algorithm

Divide-and-conquer eigenvalue algorithm

Divide-and-conquer 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 2-by-2 SVD

The singular values of a 2-by-2 matrix can be found analytically. Let the matrix be

where

are complex numbers that parameterize the matrix, and

denote the Pauli matrices

Pauli matrices

The 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 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<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

QR decomposition

In 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 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<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 Fan

Ky Fan

Ky Fan was an American mathematician and Emeritus Professor of Mathematics at the University of California, Santa Barbara .-Biography:...

k-norm of M.

The first of the Ky Fan norms, the Ky Fan 1-norm is the same as the operator norm

Operator norm

In 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 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

But, in the matrix case, M*M^½ is a normal matrix

Normal matrix

, 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 norm

Trace class

In 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 2-norm, 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 ill-posed. 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 multi-way arrays. One decomposition decomposes a tensor into a sum of rank-1 tensors, see Candecomp-PARAFAC (CP) algorithm. The CP algorithm should not be confused with a rank-R decomposition but, for a given N, it decomposes a tensor into a sum of N rank-1 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/TuckerM

Tucker decomposition

In 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....

, M-mode SVD, multilinear SVD and sometimes referred to as a higher-order SVD (HOSVD). In addition, multilinear principal component analysis

Multilinear principal component analysis

Multilinear principal-component 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 learning

Multilinear subspace learning

Multilinear 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 low-dimensional representation with certain preferred characteristics of high-dimensional...

involves the same mathematical operations as Tucker decomposition, being used in a different context of dimensionality reduction

Dimensionality reduction

In 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 operator

Bounded operator

In 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 non-zero vectors v in X...

M on a separable Hilbert space H. Namely, for any bounded operator M, there exist a partial isometry

Partial isometry

In 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 non-negative measurable f such that

where

is the multiplication by f

Multiplication operator

In 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 L²(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 calculus

Borel functional calculus

In 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 U₁ with non trivial kernel, a suitable U₂ 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/right-singular vectors to the operator case, one needs to restrict to compact operators

Compact operator on Hilbert space

In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank 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 space

Banach space

In 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 space

Hilbert space

The 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 two-dimensional Euclidean plane and three-dimensional 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 right-singular vectors of M respectively.

Compact operators on a Hilbert space

Compact operator on Hilbert space

are the closure of finite-rank operators in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:

Theorem 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 Beltrami

Eugenio Beltrami

Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...

and Camille Jordan

Camille Jordan

Marie 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 set

Complete set of invariants

In 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 invariant

Invariant (mathematics)

In 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 Sylvester

James Joseph Sylvester

James 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 Eckart

Carl Eckart

Carl Henry Eckart was an American physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner-Eckart 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 Schmidt

Erhard Schmidt

Erhard 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 Georg-August 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 Kogbetliantz

Ervand Kogbetliantz

Ervand 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 Hestenes

Magnus Hestenes

Magnus 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 algorithm

Jacobi eigenvalue algorithm

In 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 Golub

Gene H. Golub

Gene Howard Golub , Fletcher Jones Professor of Computer Science at Stanford University, was one of the preeminent numerical analysts of his generation....

and William Kahan

William Kahan

William 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 transformation

Householder transformation

In 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 most-used today.

Singular value decomposition

Statement of the theorem

Rotation, scaling, rotation

Singular values as semiaxis of an ellipse or ellipsoid

U and V are orthonormal bases

Example

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

Pseudoinverse

Solving homogeneous linear equations

Total least squares minimization

Range, null space and rank

Low-rank matrix approximation

Separable models

Nearest orthogonal matrix

The Kabsch Algorithm

Other examples

Relation to eigenvalue decomposition

Existence

Based on the spectral theorem

Based on variational characterization

Geometric meaning

Numerical Approach

Analytic Result of 2-by-2 SVD

Reduced SVDs

Thin SVD

Compact SVD

Truncated SVD

Ky Fan norms

Hilbert–Schmidt norm

Tensor SVD

Bounded operators on Hilbert spaces

Singular values and compact operators

History

See also

Libraries that support complex and real SVD

Libraries that support real SVD

Texts and demonstrations

Songs