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RE: [Q] symbolic SVD?

• To: mathgroup at smc.vnet.net
• Subject: [mg26852] RE: [mg26809] [Q] symbolic SVD?
• From: Mikael Adlers <mikael at mathcore.com>
• Date: Thu, 25 Jan 2001 01:13:29 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

Hi,
since the SVD and eigenvalue problem is closely related it is possible to
use the
function Eigenvalues/Eigenvectors to compute the symbolic singular 
values and vectors for low dimensional matrices. (However, I should not
recommend
if since the expressions will be quite awfull.)

Let A = U S Transpose[V] be the singular value decomposition of A 
(A mxn matrix, U orthonormal mxm S diagonal mxn and V orthonormal nxn) 
then the following relation is true

Sqrt[Eigenvalues[Transpose[A].A]] = Singular values of A
Eigenvectors[Transpose[A].A] = 'almost' the right singular vectors to A (V)
Eigenvectors[A.Transpose[A]] = 'almost' the left singular vectors to A (U)

To obtain the singular vectors youll have to normalize the vector length to
1
(by e.g. U = DiagonalMatrix[1/(VectorNorm[#,2]&)/@U].U )

Hope this helps,
/Mikael


 ------------------------------------------------------------------ 
 Mikael Adlers, Ph.D.          email: mikael at mathcore.com 
 MathCore AB                   phone: +4613 32 85 07 
 Teknikringen 1F               fax:         21 27 01
 SE-583 30 Linköping, Sweden                   


> -----Original Message-----
> From: research at proton.csl.uiuc.edu 
To: mathgroup at smc.vnet.net
> [mailto:research at proton.csl.uiuc.edu]
> Sent: den 24 januari 2001 10:19
> To: mathgroup at smc.vnet.net
> Subject: [mg26852] [mg26809] [Q] symbolic SVD?
> 
> 
> 
> 
> Sorry about reposting.
> I've not received any reply yet,
> and I tried several ways, but I couldn't get a clue, either.
> So if anyone can help, that would be appreciated.
> 
> -------------------------------------------------------------
> 
> I've used 'symbolic' calculation in mathematica in many ways,
> but this time, 
> I wonder if mathematica can solve symbolic matrix operation.
> Specifically, 
> 
>    m = {{a, b},
>         {c, d},
>         {e, f}}
> 
>    SingluarValues[m]
> 
> returns
> 
>   SingularValues::"svdf": "SingularValues has received a matrix 
>           with infinite precision."
> 
> (a,b,c,d,e,f are all symbolics.) 
> I need to do several symbolic matrix operations such as 
> SingularValues.
> Is there any way to do that?
> Thank you.
>