Singular value decomposition: Difference between revisions

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{{Short description|Matrix decomposition}}

{{Use dmy dates|date=October 2020}}

{{CS1 config|mode=cs1}}

[[File:Singular-Value-Decomposition.svg|thumb|Illustration of the singular value decomposition {{math|'''UΣV'''^⁎}} of a real {{math|2 × 2}} matrix {{math|'''M'''}}.{{ubli

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where <math>r \leq \min\{m,n\}</math> is the rank of {{tmath|\mathbf M.}}

The SVD is not unique,. howeverHowever, it is always possible to choose the decomposition such that the singular values <math>\Sigma_{i i}</math> are in descending order. In this case, <math>\mathbf \Sigma</math> (but not {{tmath|\mathbf U}} and {{tmath|\mathbf V}}) is uniquely determined by {{tmath|\mathbf M.}}

The term sometimes refers to the '''compact SVD''', a similar decomposition {{tmath|\mathbf M {{=}} \mathbf{U\Sigma V}^*}} in which {{tmath|\mathbf \Sigma}} is square diagonal of size {{tmath|r \times r,}} where {{tmath|r \leq \min\{m,n\} }} is the rank of {{tmath|\mathbf M,}} and has only the non-zero singular values. In this variant, {{tmath|\mathbf U}} is an {{tmath|m \times r}} [[semi-orthogonal matrix|semi-unitary matrix]] and <math>\mathbf{V}</math> is an {{tmath|n \times r}} [[semi-orthogonal matrix|semi-unitary matrix]], such that <math>\mathbf U^* \mathbf U = \mathbf V^* \mathbf V = \mathbf I_r.</math>

Mathematical applications of the SVD include computing the [[Moore–Penrose pseudoinverse|pseudoinverse]], matrix approximation, and determining the rank, [[range of a matrix|range]], and [[kernel (matrix)|null space]] of a matrix. The SVD is also extremely useful in allmany areas of science, [[engineering]], and [[statistics]], such as [[signal processing]], [[least squares]] fitting of data, and [[process control]].

== Intuitive interpretations ==

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\end{align}</math>

This particular singular value decomposition is not unique. ChoosingFor instance, we can keep {{tmath|\mathbf U}} and {{tmath|\mathbf \Sigma}} the same, but change the last two rows of {{tmath|\mathbf V^*}} such that

<math display=block>\mathbf{V}^* = \begin{bmatrix}

\color{Violet}0 & \color{Violet}10 & \color{Violet}0-1 & \color{Violet}0 & \color{Violet}0 \\

\color{Plum}-\sqrt{0 .2}& \color{Plum}0 & \color{Plum}10 & \color{Plum}0 & \color{Plum}-\sqrt{0 .8} \\

\color{Magenta}\sqrt{0.2} & \color{Magenta}0-1 & \color{Magenta}0 & \color{Magenta}0 & \color{Magenta}\sqrt{0.8} \\

\color{Orchid}\sqrt{0.4} & \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}\sqrt{0.5} & \color{Orchid}-\sqrt{0.1} \\

\color{Purple}-\sqrt{0.4} & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.5} & \color{Purple}\sqrt{0.1}

\end{bmatrix}</math>

and get an equally valid singular value decomposition. As the matrix {{tmath|\mathbf M}} has rank 3, it has only 3 nonzero singular values. In taking the product {{tmath|\mathbf{U}\mathbf{\Sigma} \mathbf{V}^* }}, the final column of {{tmath|\mathbf U}} and the final two rows of {{tmath|\mathbf{V^*} }} are multiplied by zero, so have no effect on the matrix product, and can be replaced by any unit vectors which are orthogonal to the first three and to each-other.

is also a valid singular value decomposition.

The [[#Compact SVD|compact SVD]], {{tmath|1= \mathbf M = \mathbf{U}_r\mathbf{\Sigma}_r \mathbf{V}_r^* }}, eliminates these superfluous rows, columns, and singular values:

<math display=block>\begin{align}

\mathbf{U}_r &= \begin{bmatrix}

\color{Green}0 & \color{Blue}-1 & \color{Cyan}0 \\

\color{Green}-1 & \color{Blue}0 & \color{Cyan}0 \\

\color{Green}0 & \color{Blue}0 & \color{Cyan}0 \\

\color{Green}0 & \color{Blue}0 & \color{Cyan}-1

\end{bmatrix} \\[6pt]

\mathbf \Sigma_r &= \begin{bmatrix}

3 & 0 & 0 \\

0 & \sqrt{5} & 0 \\

0 & 0 & 2

\end{bmatrix} \\[6pt]

\mathbf{V}^*_r &= \begin{bmatrix}

\color{Violet}0 & \color{Violet}0 & \color{Violet}-1 & \color{Violet}0 &\color{Violet}0 \\

\color{Plum}-\sqrt{0.2}& \color{Plum}0 & \color{Plum}0 & \color{Plum}0 &\color{Plum}-\sqrt{0.8} \\

\color{Magenta}0 & \color{Magenta}-1 & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0

\end{bmatrix}

\end{align}</math>

== SVD and spectral decomposition ==

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

A non-negative real number {{tmath|\sigma}} is a '''[[singular value]]''' for {{tmath|\mathbf M}} if and only if there exist [[unit-length vectorsvector]]s {{tmath|\mathbf u}} in {{tmath|K^m}} and {{tmath|\mathbf v}} in {{tmath|K^n}} such that

<math display=block>\begin{align}

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In the special case of {{tmath|\mathbf M}} being a [[normal matrix]], and thus also square, the [[Spectral theorem#Finite-dimensional case|spectral theorem]] ensures that it can be [[Unitary transform|unitarily]] [[Diagonalizable matrix|diagonalized]] using a basis of [[eigenvector]]s, and thus decomposed as {{tmath|\mathbf M {{=}} \mathbf U\mathbf D\mathbf U^*}} for some unitary matrix {{tmath|\mathbf U}} and diagonal matrix {{tmath|\mathbf D}} with complex elements {{tmath|\sigma_i}} along the diagonal. When {{tmath|\mathbf M}} is [[Positive-definite matrix|positive semi-definite]], {{tmath|\sigma_i}} will be non-negative real numbers so that the decomposition {{tmath|\mathbf M {{=}} \mathbf U \mathbf D \mathbf U^*}} is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase {{tmath|e^{i\varphi} }} of each {{tmath|\sigma_i}} to either its corresponding {{tmath|\mathbf V_i}} or {{tmath|\mathbf U_i.}} The natural connection of the SVD to non-normal matrices is through the [[polar decomposition]] theorem: {{tmath|\mathbf M {{=}} \mathbf S \mathbf R,}} where {{tmath|\mathbf S {{=}} \mathbf U \mathbf\Sigma \mathbf U^*}} is positive semidefinite and normal, and {{tmath|\mathbf R {{=}} \mathbf U \mathbf V^*}} is unitary.

Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of {{tmath|\mathbf M,}} while related, differ: the eigenvalue decomposition is {{tmath|1= \mathbf M = \mathbf U \mathbf D \mathbf U^{-1},}} where {{tmath|\mathbf U}} is not necessarily unitary and {{tmath|\mathbf D}} is not necessarily positive semi-definite, while the SVD is {{tmath|1= \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^*,}} where {{tmath|\mathbf \Sigma}} is diagonal and positive semi-definite, and {{tmath|\mathbf U}} and {{tmath|\mathbf V}} are unitary matrices that are not necessarily related except through the matrix {{tmath|\mathbf M.}} While only [[defective matrix|non-defective]] square matrices have an eigenvalue decomposition, any {{tmath|m \times n}} matrix has a SVD.

==Applications of the SVD==

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===Solving homogeneous linear equations===

A set of [[homogeneous linear equation]]s can be written as {{tmath|\mathbf A \mathbf x {{=}} \mathbf 0}} for a matrix {{tmath|\mathbf A}}, vector {{tmath|\mathbf x}}, and [[zero vector]] {{tmath|\mathbf x.0}}. A typical situation is that {{tmath|\mathbf A}} is known and a non-zero {{tmath|\mathbf x}} is to be determined which satisfies the equation. Such an {{tmath|\mathbf x}} belongs to {{tmath|\mathbf A}}'s [[Kernel (matrix)|null space]] and is sometimes called a (right) null vector of {{tmath|\mathbf A.}} The vector {{tmath|\mathbf x}} can be characterized as a right-singular vector corresponding to a singular value of {{tmath|\mathbf A}} that is zero. This observation means that if {{tmath|\mathbf A}} is a [[square matrix]] and has no vanishing singular value, the equation has no non-zero {{tmath|\mathbf x}} 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 {{tmath|\mathbf x}} satisfying {{tmath|\mathbf x^* \mathbf A {{=}} \mathbf 0}} with {{tmath|\mathbf x^*}} denoting the conjugate transpose of {{tmath|\mathbf x,}} is called a left null vector of {{tmath|\mathbf A.}}

===Total least squares minimization===

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where <math>\tilde{\mathbf \Sigma}</math> is the same matrix as <math>\mathbf \Sigma</math> except that it contains only the {{tmath|r}} largest singular values (the other singular values are replaced by zero). This is known as the '''[[Low-rank approximation|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 {{harvnb|Stewart|1993}}).

=== Image compression ===

[[File:Svd compression.jpg|thumb|Singular-value decomposition (SVD) image compression of a 1996 Chevrolet Corvette photograph. The original RGB image (upper-left) is compared with rank 1, 10, and 100 reconstructions.|292x292px]]One practical consequence of the low-rank approximation given by SVD is that a [[greyscale image]] represented as an <math>m \times n</math> matrix <math>\mathbf{A}</math>, can be efficiently represented by keeping the first <math>k</math> singular values and corresponding vectors. The truncated decomposition

<math>\mathbf{A}_k = \sum_{j=1}^k \sigma_j\mathbf{u}_j \mathbf{v}_j^T </math>

gives an image with the best 2-norm error out of all rank k approximations. Thus, the task becomes finding an approximation that balances retaining perceptual fidelity with the number of vectors required to reconstruct the image. Storing <math>\mathbf{A}_k</math> requires only <math>k(n + m + 1)</math> floating-point numbers compared to <math>nm</math> integers. This same idea extends to color images by applying this operation to each channel or stacking the channels into one matrix.

Since the singular values of most natural images decay quickly, most of their variance is often captured by a small <math>k</math>. For a 1528 × 1225 greyscale image, we can achieve a relative error of <math>.7%</math> with as little as <math>k = 100</math>.<ref>{{Cite book |author1=Holmes |first=Mark |title=Introduction to Scientific Computing and Data Analysis, 2nd Ed |publisher=Springer |year=2023 |isbn=978-3-031-22429-4}}</ref> In practice, however, computing the SVD can be too computationally expensive and the resulting compression is typically less storage efficient than a specialized algorithm such as [[JPEG]].

===Separable models===

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===Nearest orthogonal matrix===

It is possible to use the SVD of a square matrix {{tmath|\mathbf A}} to determine the [[orthogonal matrix]] {{tmath|\mathbf OQ}} closest to {{tmath|\mathbf A.}} The closeness of fit is measured by the [[Frobenius norm]] of {{tmath|\mathbf OQ - \mathbf A.}} The solution is the product {{tmath|\mathbf U \mathbf V^*.}}<ref>[http://www.wou.edu/~beavers/Talks/Willamette1106.pdf The Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression]</ref> This intuitively makes sense because an orthogonal matrix would have the decomposition {{tmath|\mathbf U \mathbf I \mathbf V^*}} where {{tmath|\mathbf I}} is the identity matrix, so that if {{tmath|\mathbf A {{=}} \mathbf U \mathbf \Sigma \mathbf V^*}} then the product {{tmath|\mathbf A {{=}} \mathbf U \mathbf V^*}} amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix {{tmath|\mathbf R {{=}} \mathbf U \mathbf V^*}} of the Polar Decomposition <math>\mathbf M = \mathbf R \mathbf P = \mathbf P' \mathbf R</math> in either order of stretch and rotation, as described above.

A similar problem, with interesting applications in [[shape analysis (digital geometry)|shape analysis]], is the [[orthogonal Procrustes problem]], which consists of finding an orthogonal matrix {{tmath|\mathbf OQ}} which most closely maps {{tmath|\mathbf A}} to {{tmath|\mathbf B.}} Specifically,

<math display=block>

\mathbf{O} Q = \underset\Omega\operatorname{argmin} \|\mathbf{A}\boldsymbol{\Omega} - \mathbf{B}\|_F \quad\text{subject to}\quad \boldsymbol{\Omega}^\operatorname{T}\boldsymbol{\Omega} = \mathbf{I},

</math>

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===The Kabsch algorithm===

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

===Principal Component Analysis===

The SVD can be used to construct the principal components<ref>{{cite book |last=Hastie |first=Trevor |author2=Robert Tibshirani |author3=Jerome Friedman |title=The Elements of Statistical Learning |edition=2nd |year=2009 |publisher=Springer |___location=New York |pages=535–536 |isbn=978-0-387-84857-0}}</ref> in [[principal component analysis]] as follows:

Let <math>\mathbf{X} \in \mathbb{R}^{N \times p}</math> be a data matrix where each of the <math>N</math> rows is a (feature-wise) mean-centered observation, each of dimension <math>p</math>.

The SVD of <math>\mathbf{X}</math> is:

<math display="block">

\mathbf{X} = \mathbf{V} \boldsymbol{\Sigma} \mathbf{U}^\ast

</math>

From the same reference,<ref>{{cite book |last=Hastie |first=Trevor |author2=Robert Tibshirani |author3=Jerome Friedman |title=The Elements of Statistical Learning |edition=2nd |year=2009 |publisher=Springer |___location=New York |pages=535–536 |isbn=978-0-387-84857-0}}</ref> we see that <math>\mathbf{V} \boldsymbol{\Sigma}</math> contains the scores of the rows of <math>\mathbf{X}</math> (i.e. each observation), and <math>\mathbf{U}</math> is the matrix whose columns are principal component loading vectors.

===Signal processing===

The SVD and pseudoinverse have been successfully applied to [[signal processing]],<ref>{{cite journal |last=Sahidullah |first=Md. |author2=Kinnunen, Tomi |title=Local spectral variability features for speaker verification |journal=Digital Signal Processing |date=March 2016 |volume=50 |pages=1–11 |doi=10.1016/j.dsp.2015.10.011 |bibcode=2016DSP....50....1S |url= https://erepo.uef.fi/handle/123456789/4375}}<!--https://erepo.uef.fi/handle/123456789/4375--></ref> [[image processing]]<ref name="Mademlis2018">{{cite book |last1=Mademlis |first1=Ioannis |last2=Tefas |first2=Anastasios |last3=Pitas |first3=Ioannis |title=2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) |chapter=Regularized SVD-Based Video Frame Saliency for Unsupervised Activity Video Summarization |url=https://ieeexplore.ieee.org/document/8462274 |year=2018 |pages=2691–2695 |publisher=IEEE |doi=10.1109/ICASSP.2018.8462274 |isbn=978-1-5386-4658-8 |s2cid=52286352 |access-date=19 January 2023}}</ref> and [[big data]] (e.g., in genomic signal processing).<ref>

{{Cite journal

| author = O. Alter, P. O. Brown and D. Botstein

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Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO.<ref>{{cite journal | last1 = Setyawati | first1 = Y. | last2 = Ohme | first2 = F. | last3 = Khan | first3 = S. | year = 2019| title = Enhancing gravitational waveform model through dynamic calibration | journal = Physical Review D | volume = 99| issue =2 | pages = 024010| doi=10.1103/PhysRevD.99.024010| bibcode = 2019PhRvD..99b4010S | arxiv = 1810.07060 | s2cid = 118935941 }}</ref> SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models.

Singular value decomposition is used in [[recommender systems]] to predict people's item ratings.<ref>{{cite journalreport |last1=Sarwar |first1=Badrul |last2=Karypis |first2=George |last3=Konstan |first3=Joseph A. |author3-link=Joseph A. Konstan |last4=Riedl |first4=John T. |author4-link=John T. Riedl |name-list-style=amp |year=2000 |title=Application of Dimensionality Reduction in Recommender System – A Case Study |url=httphttps://filesapps.grouplensdtic.orgmil/sti/citations/paperstr/webKDD00.pdfADA439541 |journal= |publisher=[[University of Minnesota]]}}{{void|commenthdl=11299/215429|Fabrickator|alternatetype=Technical link:report https://conservancy.umn.edu/bitstream/handle/11299/215429/00-043.pdf}}</ref> Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines.<ref>{{cite journalarXiv |last1=Bosagh Zadeh |first1=Reza |last2=Carlsson |first2=Gunnar |title=Dimension Independent Matrix Square Using MapReduce |urlyear=https://stanford.edu/~rezab/papers/dimsum.pdf2013 |bibcodeclass=2013arXiv1304cs.1467B |year=2013DS |arxiveprint=1304.1467}}</ref>

Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease [[outbreak]] detection.<ref>{{Cite journal

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In [[astrodynamics]], the SVD and its variants are used as an option to determine suitable maneuver directions for transfer trajectory design<ref name=muralidharan2023stretching>{{Cite journal|title=Stretching directions in cislunar space: Applications for departures and transfer design|first1=Vivek|last1=Muralidharan|first2=Kathleen|last2=Howell|journal =Astrodynamics | volume = 7 | issue = 2| pages = 153–178 | date = 2023 | doi = 10.1007/s42064-022-0147-z | bibcode = 2023AsDyn...7..153M|s2cid=252637213 }}</ref> and [[orbital station-keeping]].<ref name=Muralidharan2021>{{Cite journal|title=Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits |first1=Vivek|last1=Muralidharan|first2=Kathleen|last2=Howell|journal =[[Advances in Space Research]] | volume = 69 | issue = 1| pages = 620–646 | date = 2022 | doi = 10.1016/j.asr.2021.10.028 | bibcode = 2022AdSpR..69..620M|s2cid=239490016 }}</ref>

The SVD can be used to measure the similarity between real-valued matrices.<ref name=albers2025>{{Cite journal|title=Assessing the Similarity of Real Matrices with Arbitrary Shape|first1=Jasper|last1=Albers|first2=Anno|last2=Kurth|first3=Robin|last3=Gutzen|first4=Aitor|last4=Morales-Gregorio|first5=Michael|last5=Denker|first6=Sonja|last6=Gruen|first7=Sacha|last7=van Albada|first8=Markus|last8=Diesmann|journal=PRX Life| issue = 2| article-number = 023005 | date = 2025 |volume=3 | doi = 10.1103/PRXLife.3.023005 |arxiv=2403.17687 |bibcode=2025PRXL....3b3005A }}</ref> By measuring the angles between the singular vectors, the inherent two-dimensional structure of matrices is accounted for. This method was shown to outperform [[cosine similarity]] and [[Frobenius norm]] in most cases, including brain activity measurements from [[neuroscience]] experiments.

== Proof of existence ==

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By the [[extreme value theorem]], this continuous function attains a maximum at some {{tmath|\mathbf u}} when restricted to the unit sphere <math>\{\|\mathbf x\| = 1\}.</math> By the [[Lagrange multipliers]] theorem, {{tmath|\mathbf u}} necessarily satisfies

<math display=block>\nabla \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{u} - \lambda \cdot \nabla \mathbf{u}^\operatorname{T} \mathbf{u} = \mathbf{0}</math>

for some real number {{tmath|\lambda.}} The nabla symbol, {{tmath|\nabla}}, is the [[del]] operator (differentiation with respect to {{nobr|{{tmath|\mathbf x}}).}} Using the symmetry of {{tmath|\mathbf M}} we obtain

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=== Based on variational characterization ===

{{anchor|vch}}The singular values can also be characterized as the maxima of {{tmath|\mathbf u^\mathrm{T} \mathbf M \mathbf v,}} considered as a function of {{tmath|\mathbf Uu}} and {{tmath|\mathbf Vv,}} over particular subspaces. The singular vectors are the values of {{tmath|\mathbf Uu}} and {{tmath|\mathbf Vv}} where these maxima are attained.

Let {{tmath|\mathbf M}} denote an {{tmath|m \times n}} matrix with real entries. Let {{tmath|S^{k-1} }} be the unit <math>(k-1)</math>-sphere in <math> \mathbb{R}^k </math>, and define <math>\sigma(\mathbf{u}, \mathbf{v}) = \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v},</math> <math>\mathbf{u} \in S^{m-1},</math> <math>\mathbf{v} \in S^{n-1}.</math>

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== Calculating the SVD ==

=== One-sided Jacobi algorithm ===

One-sided Jacobi algorithm is an iterative algorithm,<ref>{{cite journal|first1=P.P.M. de|last1=Rijk|title=A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer|journal=SIAM J. Sci. Stat. Comput.|volume=10|pages=359359–371|year=1989|issue=2 |doi=10.1137/0910023 }}</ref>

where a matrix is iteratively transformed into a matrix with orthogonal columns. The elementary iteration is given as a [[Jacobi rotation]],

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In applications that require an approximation to the [[Moore–Penrose inverse]] of the matrix {{tmath|\mathbf M,}} the smallest singular values of {{tmath|\mathbf M}} are of interest, which are more challenging to compute compared to the largest ones.

Truncated SVD is employed in [[latent semantic indexing]].<ref>{{cite journal | last1 = Chicco | first1 = D | last2 = Masseroli | first2 = M | year = 2015 | title = Software suite for gene and protein annotation prediction and similarity search | journal = IEEE/ACM Transactions on Computational Biology and Bioinformatics | volume = 12 | issue = 4 | pages = 837–843 | doi=10.1109/TCBB.2014.2382127 | pmid = 26357324 | bibcode = 2015ITCBB..12..837C | hdl = 11311/959408 | s2cid = 14714823 | url = https://doi.org/10.1109/TCBB.2014.2382127 | hdl-access = free }}

</ref>

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==See also==

{{columns-list|colwidth=22em|

*[[Autoencoder]]

*[[Canonical correlation]]

*[[Canonical form]]

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* {{Citation | last1 = Banerjee | first1 = Sudipto | last2 = Roy | first2 = Anindya | date = 2014 | title = Linear Algebra and Matrix Analysis for Statistics | series = Texts in Statistical Science | publisher = Chapman and Hall/CRC | edition = 1st | isbn = 978-1420095388}}

* {{Cite book | last1=Bisgard | first1 = James | year = 2021 | title = Analysis and Linear Algebra: The Singular Value Decomposition and Applications | series = Student Mathematical Library | publisher = AMS | edition = 1st | isbn = 978-1-4704-6332-8}}

* {{cite journal | last1 = Chicco | first1 = D | last2 = Masseroli | first2 = M | year = 2015 | title = Software suite for gene and protein annotation prediction and similarity search | journal = IEEE/ACM Transactions on Computational Biology and Bioinformatics | volume = 12 | issue = 4 | pages = 837–843 | doi=10.1109/TCBB.2014.2382127 | pmid = 26357324 | bibcode = 2015ITCBB..12..837C | hdl = 11311/959408 | s2cid = 14714823 | url = https://doi.org/10.1109/TCBB.2014.2382127 | hdl-access = free }}

* {{Cite book | last2=Bau III | first2=David | last1=Trefethen | first1=Lloyd N. | author1-link = Lloyd N. Trefethen | title=Numerical linear algebra | publisher=Society for Industrial and Applied Mathematics | ___location=Philadelphia | isbn=978-0-89871-361-9 | year=1997 }}

* {{Cite journal | last1=Demmel | first1=James | author1-link = James Demmel | last2=Kahan | first2=William | author2-link=William Kahan | title=Accurate singular values of bidiagonal matrices | doi=10.1137/0911052 | year=1990 | journal= SIAM Journal on Scientific and Statistical Computing| volume=11 | issue=5 | pages=873–912 | citeseerx=10.1.1.48.3740 }}