       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

> -----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?
>
>
>
>
> 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.
>

```

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