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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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