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MathGroup Archive 1995

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Strange answer from Eigensystem[]!

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg839] Strange answer from Eigensystem[]!
  • From: glandry at aol.com (GLandry)
  • Date: Tue, 25 Apr 1995 02:20:25 -0400
  • Organization: America Online, Inc. (1-800-827-6364)

I have a problem w/ the eigenvectors not being correct for
the simple nonsingular matrix A (shown below).  All 
eigenvalues seem correct.  Two eigenvectors check, 2 don't..

WHY!!??

Also, using Mathcad, I get the same eigenvalues, but two 
repeated eigenvectors (rounding off 10^-4 parts):
    {.463I,0.117-0.773I,0.354+0.54I,0.212}
    {-.463I,0.117+0.773I,0.354-0.54I,0.212}

The bizarre thing to me is these eigenvectors seem to check 
too!?

Help!  I must be missing some basic degeneracy.  Unfortunately,
so in Mma.

****************************************
Gary Landry
glandry at aol.com
eelandry at utacnvx.uta.edu
________________
The University of Texas at Arlington
Department of Electrical Engineering
Box 19016
Arlington, TX 76019
****************************************

A={{0,0,0,1},{0,0,-1,0},{0,0.21,0,0},{-.21,0,0,0}};

MatrixForm[A]

0       0       0       1
0       0       -1      0
0       0.21    0       0
-0.21   0       0       0

{val,vec}=Eigensystem[A];

(Table[{A.vec[[i]],val[[i]] vec[[i]],
        A.vec[[i]]==val[[i]] vec[[i]]},{i,4}])

             -17
{{{4.17175 10    + 0. I, 0. + 0.416598 I, 0.190909 + 0. I, 
 
    0. + 0. I}, {0. + 0. I, 0. + 0.416598 I, 0.190909 + 0. I, 
 
                   -17
    0. + 1.91173 10    I}, False}, 
 
              -17
  {{4.17175 10    + 0. I, 0. - 0.416598 I, 0.190909 + 0. I, 
 
    0. + 0. I}, {0. + 0. I, 0. - 0.416598 I, 0.190909 + 0. I, 
 
                   -17
    0. - 1.91173 10    I}, False}, 
 
                   -312
  {{0. + 1.95628 10     I, -1. + 0. I, 0. + 0. I, 0. + 0. I}, 
 
   {0. + 0. I, 0. + 0. I, 0. - 0.458258 I, 0. + 0. I}, False}, 
 
                   -312
  {{0. - 1.95628 10     I, -1. + 0. I, 0. + 0. I, 0. + 0. I}, 
 
   {0. + 0. I, 0. + 0. I, 0. + 0.458258 I, 0. + 0. I}, False}}

Chop off "little" values, then the FIRST two work as solutions....

Cvec=Chop[vec];
(Table[{A.Cvec[[i]],val[[i]] Cvec[[i]],
        A.Cvec[[i]]==val[[i]] Cvec[[i]]},{i,4}])

{{{0. + 0. I, 0. + 0.416598 I, 0.190909 + 0. I, 0. + 0. I}, 
 
   {0, 0. + 0.416598 I, 0.190909 + 0. I, 0}, True}, 
 
  {{0. + 0. I, 0. - 0.416598 I, 0.190909 + 0. I, 0. + 0. I}, 
 
   {0, 0. - 0.416598 I, 0.190909 + 0. I, 0}, True}, 
 
  {{0., -1., 0., 0.}, {0, 0, 0. - 0.458258 I, 0}, False}, 
 
  {{0., -1., 0., 0.}, {0, 0, 0. + 0.458258 I, 0}, False}}


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