generalized singular value decomposition

The generalized singular value decomposition (GSVD) is a matrix decomposition on a pair of matrices which generalizes the singular value decomposition. It was introduced by Van Loan in 1976 and later developed by Paige and Saunders, which is the version described here. In contrast to the SVD, the GSVD decomposes simultaneously a pair of matrices with the same number of columns. The SVD and the GSVD, as well as some other possible generalizations of the SVD,{{Cite book | last = Hansen | first = Per Christian | name-list-style = vanc | publisher = SIAM Monographs on Mathematical Modeling and Computation | year = 1997 | title = Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion | isbn = 0-89871-403-6 }}{{cite web | last1 = de Moor | first1 = Bart L. R. | last2 = Golub | first2 = Gene H. | name-list-style = vanc | year = 1989 | title = Generalized Singular Value Decompositions A Proposal for a Standard Nomenclauture | url = http://ftp.esat.kuleuven.be/pub/SISTA/ida/reports/89-10.pdf }}{{cite journal | last1 = de Moor | first1 = Bart L. R. | last2 = Zha| first2 = Hongyuan | name-list-style = vanc | year = 1991 | title = A tree of generalizations of the ordinary singular value decomposition| journal = Linear Algebra and Its Applications | volume = 147 | pages = 469–500 | doi = 10.1016/0024-3795(91)90243-P | doi-access = free }} are extensively used in the study of the conditioning and regularization of linear systems with respect to quadratic semi-norms. In the following, let $\backslash mathbb\{F\}\; =\; \backslash mathbb\{R\}$, or $\backslash mathbb\{F\}\; =\; \backslash mathbb\{C\}$.

=== Definition ===

The generalized singular value decomposition of matrices $A\_1\; \backslash in\; \backslash mathbb\{F\}^\{m\_1\; \backslash times\; n\}$ and $A\_2\; \backslash in\; \backslash mathbb\{F\}^\{m\_2\; \backslash times\; n\}$ is$$$$

\begin{align}

A_1 & = U_1\Sigma_1 [ W^* D, 0_D] Q^*, \\

A_2 & = U_2\Sigma_2 [ W^* D, 0_D] Q^*,

\end{align}

where

• $U\_1\; \backslash in\; \backslash mathbb\{F\}^\{m\_1\; \backslash times\; m\_1\}$ is unitary,
• $U\_2\; \backslash in\; \backslash mathbb\{F\}^\{m\_2\; \backslash times\; m\_2\}$ is unitary,
• $Q\; \backslash in\; \backslash mathbb\{F\}^\{n\; \backslash times\; n\}$ is unitary,
• $$

W \in \mathbb{F}^{k \times k}

is unitary,

D \in \mathbb{R}^{k \times k}

is real diagonal with positive diagonal, and contains the non-zero singular values of $C\; =\; \backslash begin\{bmatrix\}\; A\_1\; \backslash \backslash \; A\_2\; \backslash end\{bmatrix\}$ in decreasing order,

• $0\_D\; =\; 0\; \backslash in\; \backslash mathbb\{R\}^\{k\; \backslash times\; (n\; -\; k)\}$,
• $\backslash Sigma\_1\; =\; \backslash lceil\; I\_A,\; S\_1,\; 0\_A\; \backslash rfloor\; \backslash in\; \backslash mathbb\{R\}^\{m\_1\; \backslash times\; k\}$ is real non-negative block-diagonal, where $S\_1\; =\; \backslash lceil\; \backslash alpha\_\{r\; +\; 1\},\; \backslash dots,\; \backslash alpha\_\{r\; +\; s\}\; \backslash rfloor$ with $1\; >\; \backslash alpha\_\{r\; +\; 1\}\; \backslash ge\; \backslash cdots\; \backslash ge\; \backslash alpha\_\{r\; +\; s\}\; >\; 0$, $I\_A\; =\; I\_r$, and $0\_A\; =\; 0\; \backslash in\; \backslash mathbb\{R\}^\{(m\_1\; -\; r\; -\; s)\; \backslash times\; (k\; -\; r\; -\; s)\}$,
• $\backslash Sigma\_2\; =\; \backslash lceil\; 0\_B,\; S\_2,\; I\_B\; \backslash rfloor\; \backslash in\; \backslash mathbb\{R\}^\{m\_2\; \backslash times\; k\}$ is real non-negative block-diagonal, where $S\_2\; =\; \backslash lceil\; \backslash beta\_\{r\; +\; 1\},\; \backslash dots,\; \backslash beta\_\{r\; +\; s\}\; \backslash rfloor$ with , $I\_B\; =\; I\_\{k\; -\; r\; -\; s\}$, and $0\_B\; =\; 0\; \backslash in\; \backslash mathbb\{R\}^\{(m\_2\; -\; k\; +\; r)\; \backslash times\; r\}$,
• $\backslash Sigma\_1^*\; \backslash Sigma\_1\; =\; \backslash lceil\backslash alpha\_1^2,\; \backslash dots,\; \backslash alpha\_k^2\backslash rfloor$,
• $\backslash Sigma\_2^*\; \backslash Sigma\_2\; =\; \backslash lceil\backslash beta\_1^2,\; \backslash dots,\; \backslash beta\_k^2\backslash rfloor$,
• $\backslash Sigma\_1^*\; \backslash Sigma\_1\; +\; \backslash Sigma\_2^*\; \backslash Sigma\_2\; =\; I\_k$,
• $k\; =\; \backslash textrm\{rank\}(C)$.

We denote $\backslash alpha\_1\; =\; \backslash cdots\; =\; \backslash alpha\_r\; =\; 1$, $\backslash alpha\_\{r\; +\; s\; +\; 1\}\; =\; \backslash cdots\; =\; \backslash alpha\_k\; =\; 0$, $\backslash beta\_1\; =\; \backslash cdots\; =\; \backslash beta\_r\; =\; 0$, and $\backslash beta\_\{r\; +\; s\; +\; 1\}\; =\; \backslash cdots\; =\; \backslash beta\_k\; =\; 1$. While $\backslash Sigma\_1$ is diagonal, $\backslash Sigma\_2$ is not always diagonal, because of the leading rectangular zero matrix; instead $\backslash Sigma\_2$ is "bottom-right-diagonal".

There are many variations of the GSVD. These variations are related to the fact that it is always possible to multiply $Q^*$ from the left by $E\; E^*\; =\; I$ where $E\; \backslash in\; \backslash mathbb\{F\}^\{n\; \backslash times\; n\}$ is an arbitrary unitary matrix. We denote

• $X\; =\; ([W^*\; D,\; 0\_D]\; Q^*)^*$
• $$

X^* = [0, R] \hat{Q}^*

, where $$

R \in \mathbb{F}^{k \times k}

is upper-triangular and invertible, and $$

\hat{Q} \in \mathbb{F}^{n \times n}

is unitary. Such matrices exist by RQ-decomposition.

• $Y\; =\; W^*\; D$. Then $$

Y

is invertible.

Here are some variations of the GSVD:

• MATLAB (gsvd):$$$$

\begin{aligned}

A_1 & = U_1 \Sigma_1 X^*, \\

A_2 & = U_2 \Sigma_2 X^*.

\end{aligned}

• LAPACK (LA_GGSVD):$$$$

\begin{aligned}

A_1 & = U_1 \Sigma_1 [0, R] \hat{Q}^*, \\

A_2 & = U_2 \Sigma_2 [0, R] \hat{Q}^*.

\end{aligned}

• Simplified:$$$$

\begin{align}

A_1 & = U_1\Sigma_1 [ Y, 0_D] Q^*, \\

A_2 & = U_2\Sigma_2 [ Y, 0_D] Q^*.

\end{align}

A generalized singular value of $A\_1$ and $A\_2$ is a pair $(a,\; b)\; \backslash in\; \backslash mathbb\{R\}^2$ such that

\begin{align}

\lim_{\delta \to 0} \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) / \det(\delta I_{n - k}) & = 0, \\

a^2 + b^2 & = 1, \\

a, b & \geq 0.

\end{align}

We have

• $A\_i\; A\_j^*\; =\; U\_i\; \backslash Sigma\_i\; Y\; Y^*\; \backslash Sigma\_j^*\; U\_j^*$

By these properties we can show that the generalized singular values are exactly the pairs $(\backslash alpha\_i,\; \backslash beta\_i)$. We have$$$$

\begin{aligned}

& \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) \\

= & \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta Q Q^*) \\

= & \det\left(Q \begin{bmatrix} Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k & 0 \\ 0 & \delta I_{n - k} \end{bmatrix} Q^*\right) \\

= & \det(\delta I_{n - k}) \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k).

\end{aligned}

Therefore

:$$

\begin{aligned}

{} & \lim_{\delta \to 0} \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) / \det(\delta I_{n - k}) \\

= & \lim_{\delta \to 0} \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k) \\

= & \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y) \\

= & |\det(Y)|^2 \prod_{i = 1}^k (b^2 \alpha_i^2 - a^2 \beta_i^2).

\end{aligned}

This expression is zero exactly when $a\; =\; \backslash alpha\_i$ and $b\; =\; \backslash beta\_i$ for some $i$.

In, the generalized singular values are claimed to be those which solve $\backslash det(b^2\; A\_1^*\; A\_1\; -\; a^2\; A\_2^*\; A\_2)\; =\; 0$. However, this claim only holds when $k\; =\; n$, since otherwise the determinant is zero for every pair $(a,\; b)\; \backslash in\; \backslash mathbb\{R\}^2$; this can be seen by substituting $\backslash delta\; =\; 0$ above.

Define $E^+\; =\; E^\{-1\}$ for any invertible matrix $E\; \backslash in\; \backslash mathbb\{F\}^\{n\; \backslash times\; n\}$ , $0^+\; =\; 0^*$ for any zero matrix $0\; \backslash in\; \backslash mathbb\{F\}^\{m\; \backslash times\; n\}$, and $\backslash left\backslash lceil\; E\_1,\; E\_2\; \backslash right\backslash rfloor^+\; =\; \backslash left\backslash lceil\; E\_1^+,\; E\_2^+\; \backslash right\backslash rfloor$ for any block-diagonal matrix. Then define$$A\_i^+\; =\; Q\; \backslash begin\{bmatrix\}\; Y^\{-1\}\; \backslash \backslash \; 0\; \backslash end\{bmatrix\}\; \backslash Sigma\_i^+\; U\_i^*$$It can be shown that $A\_i^+$ as defined here is a generalized inverse of $A\_i$; in particular a $\backslash \{1,\; 2,\; 3\backslash \}$-inverse of $A\_i$. Since it does not in general satisfy $(A\_i^+\; A\_i)^*\; =\; A\_i^+\; A\_i$, this is not the Moore–Penrose inverse; otherwise we could derive $(AB)^+\; =\; B^+\; A^+$ for any choice of matrices, which only holds for certain class of matrices.

Suppose , where $Q\_1\; \backslash in\; \backslash mathbb\{F\}^\{n\; \backslash times\; k\}$ and $Q\_2\; \backslash in\; \backslash mathbb\{F\}^\{n\; \backslash times\; (n\; -\; k)\}$. This generalized inverse has the following properties:

• $\backslash Sigma\_1^+\; =\; \backslash lceil\; I\_A,\; S\_1^\{-1\},\; 0\_A^T\; \backslash rfloor$
• $\backslash Sigma\_2^+\; =\; \backslash lceil\; 0^T\_B,\; S\_2^\{-1\},\; I\_B\; \backslash rfloor$
• $\backslash Sigma\_1\; \backslash Sigma\_1^+\; =\; \backslash lceil\; I,\; I,\; 0\; \backslash rfloor$
• $\backslash Sigma\_2\; \backslash Sigma\_2^+\; =\; \backslash lceil\; 0,\; I,\; I\; \backslash rfloor$
• $\backslash Sigma\_1\; \backslash Sigma\_2^+\; =\; \backslash lceil\; 0,\; S\_1\; S\_2^\{-1\},\; 0\; \backslash rfloor$
• $\backslash Sigma\_1^+\; \backslash Sigma\_2\; =\; \backslash lceil\; 0,\; S\_1^\{-1\}\; S\_2,\; 0\; \backslash rfloor$
• $A\_i\; A\_j^+\; =\; U\_i\; \backslash Sigma\_i\; \backslash Sigma\_j^+\; U\_j^*$

A generalized singular ratio of $A\_1$ and $A\_2$ is $\backslash sigma\_i=\backslash alpha\_i\; \backslash beta\_i^+$. By the above properties, $A\_1\; A\_2^+\; =\; U\_1\; \backslash Sigma\_1\; \backslash Sigma\_2^+\; U\_2^*$. Note that $\backslash Sigma\_1\; \backslash Sigma\_2^+\; =\; \backslash lceil\; 0,\; S\_1\; S\_2^\{-1\},\; 0\; \backslash rfloor$ is diagonal, and that, ignoring the leading zeros, contains the singular ratios in decreasing order. If $A\_2$ is invertible, then $\backslash Sigma\_1\; \backslash Sigma\_2^+$ has no leading zeros, and the generalized singular ratios are the singular values, and $U\_1$ and $U\_2$ are the matrices of singular vectors, of the matrix $A\_1\; A\_2^+\; =\; A\_1\; A\_2^\{-1\}$. In fact, computing the SVD of $A\_1\; A\_2^\{-1\}$ is one of the motivations for the GSVD, as "forming $AB^\{-1\}$ and finding its SVD can lead to unnecessary and large numerical errors when $B$ is ill-conditioned for solution of equations". Hence the sometimes used name "quotient SVD", although this is not the only reason for using GSVD. If $A\_2$ is not invertible, then $U\_1\; \backslash Sigma\_1\; \backslash Sigma\_2^+\; U\_2^*$is still the SVD of $A\_1\; A\_2^+$ if we relax the requirement of having the singular values in decreasing order. Alternatively, a decreasing order SVD can be found by moving the leading zeros to the back: $U\_1\; \backslash Sigma\_1\; \backslash Sigma\_2^+\; U\_2^*\; =\; (U\_1\; P\_1)\; P\_1^*\; \backslash Sigma\_1\; \backslash Sigma\_2^+\; P\_2\; (P\_2^*\; U\_2^*)$, where $P\_1$ and $P\_2$ are appropriate permutation matrices. Since rank equals the number of non-zero singular values, $\backslash mathrm\{rank\}(A\_1\; A\_2^+)=s$.

Let

• $C\; =\; P\; \backslash lceil\; D,\; 0\; \backslash rfloor\; Q^*$ be the SVD of $C\; =\; \backslash begin\{bmatrix\}\; A\_1\; \backslash \backslash \; A\_2\; \backslash end\{bmatrix\}$, where $P\; \backslash in\; \backslash mathbb\{F\}^\{(m\_1\; +\; m\_2)\; \backslash times\; (m\_1\; \backslash times\; m\_2)\}$ is unitary, and $Q$ and $D$ are as described,
• $P\; =\; [P\_1,\; P\_2]$, where $P\_1\; \backslash in\; \backslash mathbb\{F\}^\{(m\_1\; +\; m\_2)\; \backslash times\; k\}$ and $P\_2\; \backslash in\; \backslash mathbb\{F\}^\{(m\_1\; +\; m\_2)\; \backslash times\; (n\; -\; k)\}$,
• $P\_1\; =\; \backslash begin\{bmatrix\}\; P\_\{11\}\; \backslash \backslash \; P\_\{21\}\; \backslash end\{bmatrix\}$, where $P\_\{11\}\; \backslash in\; \backslash mathbb\{F\}^\{m\_1\; \backslash times\; k\}$ and $P\_\{21\}\; \backslash in\; \backslash mathbb\{F\}^\{m\_2\; \backslash times\; k\}$,
• $P\_\{11\}\; =\; U\_1\; \backslash Sigma\_1\; W^*$ by the SVD of $P\_\{11\}$, where $U\_1$, $\backslash Sigma\_1$ and $W$ are as described,
• $P\_\{21\}\; W\; =\; U\_2\; \backslash Sigma\_2$ by a decomposition similar to a QR-decomposition, where $U\_2$ and $\backslash Sigma\_2$ are as described.

Then$$\backslash begin\{aligned\}$$

C & = P \lceil D, 0 \rfloor Q^* \\

{} & = [P_1 D, 0] Q^* \\

{} & = \begin{bmatrix} U_1 \Sigma_1 W^* D & 0 \\ U_2 \Sigma_2 W^* D & 0 \end{bmatrix} Q^* \\

{} & = \begin{bmatrix} U_1 \Sigma_1 [W^* D, 0] Q^* \\ U_2 \Sigma_2 [W^* D, 0] Q^* \end{bmatrix} .

\end{aligned}We also haveThereforeSince $P\_1$ has orthonormal columns, $||P\_1||\_2\; \backslash leq\; 1$. Therefore$$||\backslash Sigma\_1||\_2\; =\; ||U\_1^*\; P\_1\; W||\_2\; =\; ||P\_1||\_2\; \backslash leq\; 1.$$We also have for each $x\; \backslash in\; \backslash mathbb\{R\}^k$ such that $||x||\_2\; =\; 1$ that$$||P\_\{21\}\; x||\_2^2\; \backslash leq\; ||P\_\{11\}\; x||\_2^2\; +\; ||P\_\{21\}\; x||\_2^2\; =\; ||P\_\{1\}\; x||\_2^2\; \backslash leq\; 1.$$Therefore $||P\_\{21\}||\_2\; \backslash leq\; 1$, and$$||\backslash Sigma\_2||\_2\; =\; ||\; U\_2^*\; P\_\{21\}\; W\; ||\_2\; =\; ||P\_\{21\}||\_2\; \backslash leq\; 1.$$

File:Tensor Generalized Singular Value Decomposition following et int. Alter PLoS One 2015 and Alter NCI Physical Sciences in Oncology 2015.jpg

The GSVD, formulated as a comparative spectral decomposition,{{cite journal | vauthors = Alter O, Brown PO, Botstein D | title = Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different organisms | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 100 | issue = 6 | pages = 3351–6 | date = March 2003 | pmid = 12631705 | pmc = 152296 | doi = 10.1073/pnas.0530258100 | bibcode = 2003PNAS..100.3351A | doi-access = free }} has been successfully applied to signal processing and data science, e.g., in genomic signal processing.{{cite journal | vauthors = Lee CH, Alpert BO, Sankaranarayanan P, Alter O | title = GSVD comparison of patient-matched normal and tumor aCGH profiles reveals global copy-number alterations predicting glioblastoma multiforme survival | journal = PLOS ONE| volume = 7 | issue = 1 | pages = e30098 | date = January 2012 | pmid = 22291905 | pmc = 3264559 | doi = 10.1371/journal.pone.0030098 | bibcode = 2012PLoSO...730098L | doi-access = free }}{{cite journal | vauthors = Aiello KA, Ponnapalli SP, Alter O | title = Mathematically universal and biologically consistent astrocytoma genotype encodes for transformation and predicts survival phenotype | journal = APL Bioengineering | volume = 2 | issue = 3 | pages = 031909 | date = September 2018 | pmid = 30397684 | pmc = 6215493 | doi = 10.1063/1.5037882 }}{{cite journal | vauthors = Ponnapalli SP, Bradley MW, Devine K, Bowen J, Coppens SE, Leraas KM, Milash BA, Li F, Luo H, Qiu S, Wu K, Yang H, Wittwer CT, Palmer CA, Jensen RL, Gastier-Foster JM, Hanson HA, Barnholtz-Sloan JS, Alter O | title = Retrospective Clinical Trial Experimentally Validates Glioblastoma Genome-Wide Pattern of DNA Copy-Number Alterations Predictor of Survival | journal = APL Bioengineering | volume = 4 | issue = 2 | pages = 026106 | date = May 2020 | doi = 10.1063/1.5142559 | pmid = 32478280 | pmc = 7229984 | id = [https://www.eurekalert.org/pub_releases/2020-05/uouh-gpf051320.php Press Release] | doi-access = free }}

These applications inspired several additional comparative spectral decompositions, i.e., the higher-order GSVD (HO GSVD) and the tensor GSVD.

It has equally found applications to estimate the spectral decompositions of linear operators when the eigenfunctions are parameterized with a linear model, i.e. a reproducing kernel Hilbert space.{{Cite arXiv|last1=Cabannes|first1=Vivien|last2=Pillaud-Vivien|first2=Loucas|last3=Bach|first3=Francis|last4=Rudi|first4=Alessandro|date=2021|title=Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning|class=stat.ML|eprint=2009.04324}}

The weighted version of the generalized singular value decomposition (GSVD) is a constrained matrix decomposition with constraints imposed on the left and right singular vectors of the singular value decomposition.{{cite book | vauthors = Jolliffe IT | title = Principal Component Analysis | url = https://archive.org/details/principalcompone00joll_0 | series = Springer Series in Statistics | edition = 2nd | publisher = Springer | location = NY | date = 2002 | isbn = 978-0-387-95442-4 | url-access = registration }}

{{Cite book | last = Greenacre | first = Michael | name-list-style = vanc | publisher = Academic Press | location = London | year = 1983 | title = Theory and Applications of Correspondence Analysis | isbn = 978-0-12-299050-2 }}{{Cite journal| vauthors = Abdi H, Williams LJ |year = 2010 | title = Principal component analysis. | journal = Wiley Interdisciplinary Reviews: Computational Statistics | volume = 2 |issue=4 | pages = 433–459 | doi=10.1002/wics.101|s2cid = 122379222 }} This form of the GSVD is an extension of the SVD as such. Given the SVD of an m×n real or complex matrix M

:$M\; =\; U\backslash Sigma\; V^*\; \backslash ,$

where

:$U^*\; W\_u\; U\; =\; V^*\; W\_v\; V\; =\; I.$

Where I is the identity matrix and where $U$ and $V$ are orthonormal given their constraints ($W\_u$ and $W\_v$). Additionally, $W\_u$ and $W\_v$ are positive definite matrices (often diagonal matrices of weights). This form of the GSVD is the core of certain techniques, such as generalized principal component analysis and Correspondence analysis.

The weighted form of the GSVD is called as such because, with the correct selection of weights, it generalizes many techniques (such as multidimensional scaling and linear discriminant analysis).{{cite book | vauthors = Abdi H | date = 2007 | chapter = Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD). | veditors = Salkind NJ | title = Encyclopedia of Measurement and Statistics. | url = https://archive.org/details/encyclopediameas00salk | url-access = limited | location = Thousand Oaks (CA) | publisher = Sage | pages = [https://archive.org/details/encyclopediameas00salk/page/n939 907]–912 }}

{{Reflist|refs=

{{cite journal | vauthors = Bradley MW, Aiello KA, Ponnapalli SP, Hanson HA, Alter O | title = GSVD- and tensor GSVD-uncovered patterns of DNA copy-number alterations predict adenocarcinomas survival in general and in response to platinum | journal = APL Bioengineering | volume = 3 | issue = 3 | pages = 036104 | date = September 2019 | pmid = 31463421 | pmc = 6701977 | doi = 10.1063/1.5099268 | id = [https://alterlab.org/publications/Bradley_et_al_APL_Bioeng_2019_Supplementary_Material.pdf Supplementary Material] }}

{{cite journal | vauthors = Sankaranarayanan P, Schomay TE, Aiello KA, Alter O | title = Tensor GSVD of patient- and platform-matched tumor and normal DNA copy-number profiles uncovers chromosome arm-wide patterns of tumor-exclusive platform-consistent alterations encoding for cell transformation and predicting ovarian cancer survival | journal = PLOS ONE| volume = 10 | issue = 4 | pages = e0121396 | date = April 2015 | pmid = 25875127 | pmc = 4398562 | doi = 10.1371/journal.pone.0121396 | bibcode = 2015PLoSO..1021396S | doi-access = free }}

{{cite journal | last= Van Loan | first = Charles F. | name-list-style = vanc | year = 1976 | title = Generalizing the Singular Value Decomposition | journal = SIAM J. Numer. Anal. | volume = 13 | issue = 1| pages = 76–83 | doi = 10.1137/0713009 | bibcode = 1976SJNA...13...76V }}

{{cite journal | last1 = Paige | first1 = C. C. | last2 = Saunders | first2 = M. A. | name-list-style = vanc | year = 1981 | title = Towards a Generalized Singular Value Decomposition | journal = SIAM J. Numer. Anal. | volume = 18 | issue = 3| pages = 398–405| doi = 10.1137/0718026 | bibcode = 1981SJNA...18..398P }}

{{cite journal | vauthors = Ponnapalli SP, Saunders MA, Van Loan CF, Alter O | title = A higher-order generalized singular value decomposition for comparison of global mRNA expression from multiple organisms | journal = PLOS ONE| volume = 6 | issue = 12 | pages = e28072 | date = December 2011 | pmid = 22216090 | pmc = 3245232 | doi = 10.1371/journal.pone.0028072 | bibcode = 2011PLoSO...628072P | doi-access = free }}

}}

{{refbegin}}

• {{Cite book | last1 = Golub | first1 = Gene | last2 = Van Loan | first2 = Charles | name-list-style = vanc | publisher = Johns Hopkins University Press | location = Baltimore | year = 1996 | title = Matrix Computation | edition = Third | isbn = 0-8018-5414-8 }}
• LAPACK manual [http://www.netlib.org/lapack/lug/node36.html]

{{refend}}

Category:Linear algebra

Category:Singular value decomposition