SingularValueDecomposition

gives the singular value decomposition for a numerical matrix m as a list of matrices {u,σ,v}, where σ is a diagonal matrix and m can be written as u.σ.ConjugateTranspose[v].

SingularValueDecomposition[{m,a}]

gives the generalized singular value decomposition of m with respect to a.

gives the singular value decomposition associated with the k largest singular values of m.

gives the decomposition for the k largest singular values, or as many as are available.

Details and Options

• The matrix m may be rectangular.
• The diagonal elements of σ are the singular values of m.
• SingularValueDecomposition sets to zero any singular values that would be dropped by SingularValueList.
• The option Tolerance can be used as in SingularValueList to determine which singular values will be considered to be zero. »
• u and v are column orthonormal matrices, whose transposes can be considered as lists of orthonormal vectors.
• SingularValueDecomposition[{m,a}] gives a list of matrices {{u,ua},{w,wa},v} such that m can be written as u.w.ConjugateTranspose[v] and a can be written as ua.wa.ConjugateTranspose[v]. »
• gives the decomposition for the k largest singular values, or as many as are available.
• With the setting TargetStructure->"Structured", returns the matrices {u,σ,v} as structured matrices.

Examples

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Basic Examples(3)

Compute a singular value decomposition:

Reconstruct the input matrix:

Compute a singular value decomposition for an invertible matrix:

The matrix of singular values is also invertible:

Reconstruct the input matrix:

Compute a singular value decomposition for an invertible matrix:

Format the results:

Reconstruct the input matrix:

Scope(18)

Basic Uses(7)

Find the singular value decomposition of a machine-precision matrix:

Format the result:

Singular value decomposition of a complex matrix:

Singular value decomposition for an exact matrix:

Singular value decomposition for an arbitrary-precision matrix:

Singular value decomposition of a symbolic matrix:

The singular value decomposition of a large numerical matrix is computed efficiently:

Singular value decomposition of a non-square matrix:

Subsets of Singular Values(5)

Find the singular value decomposition associated with the three largest singular values of a matrix:

Unlike the full decomposition, these matrices do not recreate any part of the matrix exactly:

Find singular value decomposition associated with the three smallest singular values:

Find the "compact" decomposition associated with the nonzero singular values:

This decomposition still has sufficient information to reconstruct the matrix:

The full singular value decomposition contains a row of zeros:

Find the "thin" decomposition of a non-rectangular matrix:

The matrix is square:

This decomposition still has sufficient information to reconstruct the matrix:

The full singular value decomposition contains rows or columns of zeros in a rectangular :

Find the decomposition associated with the three largest singular values, or as many as there are if fewer:

Compute a truncated singular value decomposition for a matrix with repeated singular values:

Repeated singular values are counted separately when doing a partial decomposition:

Generalized Singular Value Decomposition(2)

Find the generalized singular value decomposition of a machine-precision real matrix:

Verify the result:

Find the generalized singular value decomposition of a machine-precision complex matrix:

Verify the result:

Special Matrices(4)

Singular value decomposition of sparse matrices:

Find the decomposition associated to the three largest singular values:

Visualize the three right-singular vectors:

Singular value decomposition of structured matrices:

Use a different structure:

The units go with the singular values:

Singular value decomposition of an identity matrix:

Verify the decomposition:

and could have been chosen to be identity matricesthe decomposition is not unique:

Singular value decomposition of HilbertMatrix:

Options(3)

Tolerance(1)

m is a nearly singular matrix:

To machine precision, the matrix is effectively singular:

With a smaller tolerance, the nonzero singular value is detected:

The default tolerance is based on precision, so the small value is detected with precision 20:

TargetStructure(2)

A real rectangular matrix:

With TargetStructure->"Dense", the result of SingularValueDecomposition is a list of three dense matrices:

With TargetStructure->"Structured", the result of SingularValueDecomposition is a list containing two OrthogonalMatrix objects and a DiagonalMatrix:

A complex rectangular matrix:

With TargetStructure->"Dense", the result of SingularValueDecomposition is a list of three dense matrices:

With TargetStructure->"Structured", the result of SingularValueDecomposition is a list containing two UnitaryMatrix objects and a DiagonalMatrix:

Applications(11)

Geometry of SVD(5)

Compute the singular value decomposition of the 2×2 matrix :

The action of is a rotation and possiblyas happens for the axis in this casea reflection:

The action of is a scalingeither a dilation or compressionalong each axis:

The action of is a rotation and possiblythough not in this casea reflection in the target space:

Compute the singular value decomposition of the 3×2 matrix :

After the rotation in the plane by the matrix, the matrix embeds the unit circle as an ellipse in 3D:

The matrix rotates the ellipse in three dimensions:

Compute the singular value decomposition of the 2×2 matrix :

Let and denote the columns, respectively, of and :

is the direction in which is maximized, and the maximum value is :

Similarly, is the direction in which is minimized, and the minimum value is :

Visualize , and the unit circle along with their image under the multiplication on the left by :

is the direction in which is maximized, and again the maximum value is :

Similarly, is the direction in which is minimized, and again the minimum value is :

Visualize , and the unit circle along with their image under the multiplication on the right by :

Compute the singular value decomposition of the 3×2 matrix :

Let and denote the columns, respectively, of and :

is the direction in which is maximized, and the maximum value is :

Similarly, is the direction in which is minimized, and the minimum value is :

Visualize , and the image of the unit circle in the plane under left-multiplication by :

is the direction in which is maximized, and again the maximum value is :

minimizes the minimum is zero, as the sphere is compressed into an ellipse in the plane:

maximizes subject to the constraint , and the maximum value is :

Visualize , , and the image of the unit sphere in the plane under right-multiplication by :

Compute the singular value decomposition of the 3×3 matrix :

Let and denote the columns, respectively, of and :

is the direction in which is maximized, and the maximum value is :

is the direction in which is maximized if , and the maximum value is :

Similarly, is the direction in which is minimized, and the minimum value is :

Visualize , , and the unit sphere along with their image under the multiplication on the left by :

is the direction in which is maximized, and again the maximum value is :

is the direction in which is maximized if , and again the maximum value is :

Similarly, is the direction in which is minimized, and again the minimum value is :

Visualize , , and the unit sphere along with their image under the multiplication on the right by :

Least Squares and Curve Fitting(6)

If the linear system has no solution, the best approximate solution is the least-squares solution. That is the solution to , where is the orthogonal projection of onto the column space of , which can be computed using the singular value decomposition. Consider the following and :

The linear system is inconsistent:

Find the matrix of the compact singular value decomposition of . Its columns are orthonormal and span :

Compute the orthogonal projection of onto the space spanned by the columns of :

Visualize , its projections onto the columns of and :

Solve :

Confirm the result using LeastSquares:

Solve the least-squares problem for the following and using only the singular value decomposition:

Compute the compact singular value decomposition where only the nonzero singular values are kept:

Let :

By definition, , so , the orthogonal projection of onto :

Thus, is the solution to the least-squares problem, as confirmed by LeastSquares:

Solve the least-squares problem for the following and two different ways: by projecting onto the column space of using just the matrix of the singular value decomposition, and the direct solution using the full decomposition. Compare and explain the results:

Compute the compact singular value decomposition of:

Compute the orthogonal projection of onto :

Solve :

The direct solution can be found using , as both and are real-valued:

While x and xPerp are different, both solve the least-squares problem because m.x==m.xPerp:

The two solutions differ by an element of NullSpace[m]:

Note that LeastSquares[m,b] gives the result using the direct method:

For the matrices and that follow, find a matrix that minimizes :

One solution, in this case unique, is given by :

This result could also have been obtained using LeastSquares[m,b]:

SingularValueDecomposition can be used to find a best-fit curve to data. Consider the following data:

Extract the and coordinates from the data:

Construct a design matrix, whose columns are and , for fitting to a line :

Get the coefficients and for a linear leastsquares fit using a thin singular value decomposition:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find the best-fit parabola to the following data:

Extract the and coordinates from the data:

Construct a design matrix, whose columns are , and , for fitting to a line :

Get the coefficients , and for a leastsquares fit:

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Properties & Relations(13)

The singular value decomposition {u,σ,v} of m decomposes m as u.σ.ConjugateTranspose[v]:

If a is an n×m matrix with decomposition {u,σ,v}, then u is an n×n matrix:

v is an m×m matrix:

And σ is an n×m matrix:

is built from the eigensystems of and :

Compute the eigensystem of :

The columns of are the eigenvectors:

Compute the eigensystem of :

The columns of are the eigenvectors:

Since has fewer rows than columns, the diagonal entries of are :

The first right singular vector can be found by maximizing over all unit vectors:

Each subsequent vector is a maximizer with the constraint that it is perpendicular to all previous vectors:

Compare the with the found by SingularValueDecomposition; they are the same up to sign:

The analogous statement holds for the left singular vectors with :

The diagonal entries of are the respective maximum values:

If is the smaller of the dimensions of , the first columns of and are related by :

The first columns of and are also related by :

If m is a square matrix, the product of the diagonal elements of equals Abs[Det[m]]:

If is a normal matrix, both and are composed of the same vectors:

The vectors will appear in a different order unless is positive semidefinite and Hermitian:

The diagonal entries of equal Abs[Eigenvalues[m]]:

For positive definite and Hermitian , SingularValueDecomposition and Eigensystem coincide:

and are equal:

Their columns are unit eigenvectors of :

The nonzero elements of are the eigenvalues of :

MatrixRank[m] equals the number of nonzero singular values:

The compact decomposition that only keeps nonzero singular values can compute :

If decomposes as , then :

A matrix m that is an outer product of two vectors has MatrixRank[m]==1:

The nonzero singular value of m is the product of the norms of the vectors:

The corresponding left and right singular vectors are the input vectors, normalized:

SingularValueDecomposition[{m,a}] decomposes m as u.w.ConjugateTranspose[v]:

It decomposes a as ua.wa.ConjugateTranspose[v]:

SingularValueDecomposition[{m,a}] can be related to Eigensystem[{m.m,a.a}]:

The diagonal elements of w are /:

The diagonal elements of wa are 1/:

The columns of v are scaled multiples of the columns of Conjugate[Inverse[vλ]]:

The magnitude of the scaling is the ratio of the corresponding diagonal elements of w and vλ.m.m.vλ:

Equivalently, it is the ratio of the corresponding diagonal elements of wa and vλ.ma.ma.vλ:

Possible Issues(1)

m is a 2×1000 random matrix:

The full singular value decomposition is very large because u is a 1000×1000 matrix:

The condensed singular value decomposition is much smaller:

It still contains sufficient information to reconstruct m:

Wolfram Research (2003), SingularValueDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/SingularValueDecomposition.html (updated 2024).

Text

Wolfram Research (2003), SingularValueDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/SingularValueDecomposition.html (updated 2024).

CMS

Wolfram Language. 2003. "SingularValueDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/SingularValueDecomposition.html.

APA

Wolfram Language. (2003). SingularValueDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SingularValueDecomposition.html

BibTeX

@misc{reference.wolfram_2024_singularvaluedecomposition, author="Wolfram Research", title="{SingularValueDecomposition}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/SingularValueDecomposition.html}", note=[Accessed: 17-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_singularvaluedecomposition, organization={Wolfram Research}, title={SingularValueDecomposition}, year={2024}, url={https://reference.wolfram.com/language/ref/SingularValueDecomposition.html}, note=[Accessed: 17-September-2024 ]}