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---
title: mtx_svd
description: singular value decomposition of a matrix
categories:
- object
pdcategory: Matrix Transformation
see_also:
- mtx_qr
inlets:
1st:
- type: matrix
description: \(n\times m\) matrix \(\boldsymbol A\) to decompose
outlets:
1st:
- type: matrix
description: \(n\times n\) matrix of left-singular vectors \(\boldsymbol U\)
1:
- type: list
description: list of singular values \(\boldsymbol s\)
2:
- type: matrix
description: \(m\times m\) matrix of right-singular vectors \(\boldsymbol V\)
draft: false
---
This object depends on the GNU Scientific Library. The matrices lists involved have the properties:
$$ \boldsymbol A_{n\times m} = \boldsymbol U \mathrm{diag}_{n\times m}\\{\boldsymbol s\\} \boldsymbol{V}^\intercal $$
$$ \boldsymbol U\boldsymbol U^\intercal=\boldsymbol U^\intercal\boldsymbol U=\boldsymbol I_{n\times n} $$
$$ \boldsymbol V\boldsymbol V^\intercal=\boldsymbol V^\intercal\boldsymbol V=\boldsymbol I_{m\times m} $$
$$ \boldsymbol s_{\mathrm{min}\\{n,m\\}\times 1}=[s_1,s_2,\dots]^\intercal $$ with $$ s_1\geq s_2\geq s_3 \dots \geq 0 $$
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