*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 ____________________________________________________________________