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MathGroup Archive 2005

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Re: Quadratic Form Contours

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57756] Re: Quadratic Form Contours
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 7 Jun 2005 05:59:45 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <d7pa71$sqt$1@smc.vnet.net> <d83dqf$o1e$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <d83dqf$o1e$1 at smc.vnet.net>, dh <dh at metrohm.ch> wrote:

> To use "Ellipsoid" we need the semi-length and directions of the axes.
> The direction we obtain from: Eigenvectors[A]
> the half-length from: Sqrt[ b/Eigenvalues[A]]
> Therefore, an ellipsoid would be specified by:
> 
> Ellipsoid[{x,y}, Sqrt[ b/Eigenvalues[A]], Eigenvectors[A] ]

Not quite. If A is an exact matrix then Eigenvectors[A] will not be 
normalised. Also, you need to transpose the Eigenvectors. The following 
code does the job:

  << Statistics`MultiDescriptiveStatistics`

 MatrixToEllipsoid[m_, b_, origin_:{0,0}] := 
 Module[{a = Transpose[Eigenvectors[m]], r = Sqrt[b/Eigenvalues[m]]}, 
    Ellipsoid[origin, r, a/(Norm /@ a)]]

For example, with

  A = {{5, -2}, {-2, 8}}; b = 36;

the ellipsoid equation is obtained via

 eqn = ExpandAll[{x, y} . A . {x, y} == b]

We can plot the contours of this as follows:

  ContourPlot[First[eqn], {x, -4, 4}, {y, -4, 4}, 
   Contours -> {b}, ContourShading -> False]; 

Alternatively, we can visualize the ellipse using

 Show[Graphics[MatrixToEllipsoid[A,b]], 
    AspectRatio -> Automatic, Axes -> True]

The two pictures agree.

Cheers,
Paul

> The erason for this is:
> If you turn the coordinate system in direction of the Eigenvectors, the 
> quadratic form looks like:
> 
> x1^2/ew1 + x2^2/ew2 == b
> 
> where ew means eigenvalue.
> sincerely, Daniel
> 
> Joerg Schaber wrote:
> > Hi,
> > 
> > does anybody know a simple way to calculate 2-D ellipsoids in x={x1,x2} 
> > of a quadatric form solving xAx=b for a graphical output, i.e. contour 
> > lines of the quadrtic form expressed as the Graphics primitive Ellipsoid?
> > I suppose that the option ParameterConfidenceRegion of NonlinearRegress 
> > does something like that, but how?
> > 
> > best,
> > 
> > joerg
> >

-- 
Paul Abbott                                      Phone: +61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul
        http://InternationalMathematicaSymposium.org/IMS2005/


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