       Re: Re: slow eigenvectors and eigenvalues

• To: mathgroup at smc.vnet.net
• Subject: [mg52991] Re: [mg52974] Re: slow eigenvectors and eigenvalues
• From: "Sungjin Kim" <kimsj at mobile.snu.ac.kr>
• Date: Sun, 19 Dec 2004 06:14:33 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```> The eigenvectors of a Toeplitz matrix (a very large one, so that the edge
> effects can be ignored) is made up of sine waves.
Or, it will be made up of 'exp(j*w)' once complex values are involved to the
input signal. If the size of the Toeplitz matrix can be set-up as 2^n by
2^n, you may get the results even faster using FFT and DCT for complex and
real values, respectively.

Best regards,
- James Sungjin Kim
communication at SAMSUNG.com
kimsj at mobile.SNU.ac.kr

-----Original Message-----
From: Steve Luttrell [mailto:steve_usenet at _removemefirst_luttrell.org.uk]
To: mathgroup at smc.vnet.net
Subject: [mg52991] [mg52974] Re: slow eigenvectors and eigenvalues

The eigenvectors of a Toeplitz matrix (a very large one, so that the edge
effects can be ignored) is made up of sine waves. This is because the left
hand side of the eigenequation T.x=a x is a convolution, so if you transform

to Fourier space it is a multiplication (i.e. the convolution theorem),
which is a diagonal operation whose eigenvectors are the individual Fourier
components.

Steve Luttrell

"Janusz" <janus at sci.pam.szczecin.pl> wrote in message
news:cpueev\$gc5\$1 at smc.vnet.net...
> Hello!
>
> I need eigenvectors and eigenvalues of Toeplitz matrix 1000 by 1000 or
> greater. These are values of autocorrelation of processed signal. My
> Version 5. does it for a long time. After a few hours I aborted
> calculations. What should I do to make it faster?
>
> Thanks in Advance
> Regards
>
> Janusz Kowalski
> Szczecin, Poland
>

```

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