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MathGroup Archive 2000

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SingularValues of symmetric real matrix not converging

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25759] SingularValues of symmetric real matrix not converging
  • From: Albert Maydeu-Olivares <amaydeu at tinet.fut.es>
  • Date: Sat, 21 Oct 2000 14:43:12 -0400 (EDT)
  • Organization: Universitat de Barcelona
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

I'm not very good at numerical methods. Consider the following 8 x 8
real matrix

mat = {{0.1618531249999996, -0.023121875000000416, -0.02312187500000039,
\
-0.02312187500000043, -0.023121875000000416, -0.02312187500000041, \
-0.02312187500000041, -0.023121875000000368}, {-0.023121875000000416, 
        0.1618531249999996, -0.02312187500000043, -0.023121875000000458,
\
-0.023121875000000458, -0.023121875000000465, -0.023121875000000465, \
-0.02312187500000045}, {-0.023121875000000416, -0.02312187500000043, 
        0.1618531249999996, -0.02312187500000039, -0.02312187500000039,
\
-0.023121875000000396, -0.023121875000000396, -0.023121875000000396}, \
{-0.023121875000000375, -0.023121875000000375, -0.02312187500000039, 
        0.1618531249999996, -0.02312187500000039, -0.02312187500000039,
\
-0.023121875000000396, -0.02312187500000039}, {-0.02312187500000036, \
-0.023121875000000375, -0.02312187500000039, -0.02312187500000039, 
        0.1618531249999996, -0.02312187500000039, -0.02312187500000039,
\
-0.02312187500000039}, {-0.023121875000000375, -0.02312187500000038, \
-0.02312187500000039, -0.02312187500000039, -0.02312187500000039, 
        0.1618531249999996, -0.023121875000000392,
-0.023121875000000392}, \
{-0.023121875000000357, -0.023121875000000364, -0.023121875000000378, \
-0.023121875000000385, -0.023121875000000385, -0.023121875000000392, 
        0.1618531249999996, -0.023121875000000385},
{-0.02312187500000036, \
-0.023121875000000375, -0.02312187500000038, -0.02312187500000038, \
-0.023121875000000378, -0.023121875000000385, -0.023121875000000385, 
        0.1618531249999996}};

One can readily verify that it's perfectly symmetric. Its eigenvalues
are

{0.18497500000000006`, 0.184975`, 0.18497499999999997`, 
    0.18497499999999992`, 0.1849749999999999`, 0.18497499999999992`, 
    0.1849749999999999`, \(-3.164135620181696`*^-15\)}

How, when I apply SingularValues it fails to converge? Can I specify a
Tolerance to have it converge?

I have one further question. How can I use its eigenvalues and
eigenvectors to reconstruct it? I know I can do that from the normalized
eigenvectors and the eigenvalues but the eigenvectors are not
normalized. Can anyone suggest me how to normalize the eigenvectors and
re-construct the matrix?

One thing that puzzles me is that the eigenvectors I obtain usin
Eigensystem[mat] and Eigenvectors[mat] are different. Why?

Thanks for your help.

Albert


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