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Physical applications of matrix congruence?

  1. Jan 5, 2010 #1
    Just curious, are there any?
  2. jcsd
  3. Jan 5, 2010 #2


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    What do you mean by "matrix congruence"? Congruent in what sense?
  4. Jan 5, 2010 #3
  5. Jan 5, 2010 #4
    Consider, for example, the diagonalization of the Moment of Inertia tensor, to find the principal moments of inertia and the respective principal axis of rotation.
  6. Jan 5, 2010 #5
    But in that case similarity transformation would be enough, it happens to be a congruence transformation just because inertia tensor is real symmetric. So nothing nontrivial about congruence relation is actually invoked.
  7. Jan 6, 2010 #6
    Yes, but the fact that this particular tensor (and many other 2nd order ones) is symmetric is a nontrivial fact, a consequence of deeper physical laws. What kind of application would you consider nontrivial?
  8. Jan 6, 2010 #7
    In the inertia tensor case we only need to treat it as similarity transformation, it'll work just fine. No special properties of a congruence transformation are used, so I don't think it's a application of congruence transformation.
  9. Jan 6, 2010 #8
    I disagree: a similarity transformation does not have to preserve the orthogonality of the reference frame axis', while a congruence (which is a particular case) does. The existence of the principal axis of rotation and moments of inertia depends on that preservation property.
  10. Jan 6, 2010 #9
    As you said, only in a particular case congruence preserve the orthogonality, then why not just view it as a particular case of similarity transformation? It's not fair you compare the special case of congruence with the general case of similarity.
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