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MathGroup Archive 1998

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Re: question


  • To: mathgroup@smc.vnet.net
  • Subject: [mg10324] Re: question
  • From: Paul Abbott <paul@physics.uwa.edu.au>
  • Date: Thu, 8 Jan 1998 23:40:49 -0500
  • Organization: University of Western Australia
  • References: <687po8$1jc@smc.vnet.net> <68s9f8$7o0$2@dragonfly.wolfram.com>

Daniel Lichtblau wrote:

> In article <687po8$1jc@smc.vnet.net> "=?big5?B?p0WuYbvK?="
> <g8673007@cc.nchulc.edu.tw> writes:
> >
> > I found that Mathematica may deal poorly with the orthogonal matrix
> > problem. Just as below shows, I can not solve my problem using
> > Mathematica. Maybe you have right way to finish it, I think and hope
> > so. If it is true, excuse me, I would thank you to let me know the
> > answer.
> > Following is my input and output in Mathematica: a={{1,2},{3,4}}
> > {{1, 2}, {3, 4}}
> > p={{a11,a12},{a21,a22}}
> > {{a11, a12}, {a21, a22}}
> > ans=Solve[Transpose[p].a.p==IdentityMatrix[2]] {}

> It is correct that the solution set is empty.
> 
> Note that the right-hand-side of the matrix equation is symmetric while
> the left-hand-side is not (which is a good indication that the
> equations   are inconsistent, because now the system is
> overdetermined).

This is certainly true.  However, the user effectively wants to
diagonalize the matrix a and this can be done as follows:

In[1]:= a={{1,2},{3,4}};

The matrix p can be found as follows:

In[2]:= p=Transpose[Eigenvectors[a]]

Out[2]=
  1                  1
{{- (-3 - Sqrt[33]), - (-3 + Sqrt[33])}, {1, 1}}
  6                  6

p does diagonalize the matrix a (note the use of Inverse instead of
Transpose):

In[3]:= Simplify[Inverse[p].a.p]
Out[3]=
  1                         1
{{- (5 - Sqrt[33]), 0}, {0, - (5 + Sqrt[33])}}
  2                         2

and the diagonal elements do agree with the eigenvalues:

In[4]:= Eigenvalues[a]
Out[4]=
 1                 1
{- (5 - Sqrt[33]), - (5 + Sqrt[33])}
 2                 2
	
Cheers,
	Paul 

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul@physics.uwa.edu.au  AUSTRALIA                            
http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________



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