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I Third Invariant expressed with Cayley-Hamilton Theorem

  1. Apr 11, 2016 #1
    The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at "Replacing the values of IT and IIT with those in 1.269. Could someone please explain? Thanks

    upload_2016-4-11_9-14-6.png upload_2016-4-11_9-17-24.png upload_2016-4-11_9-17-35.png
    with 1.269 being:
  2. jcsd
  3. Apr 11, 2016 #2


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    We need to show that the RHS of the equations in the first two boxes are equal. To minimise the latex coding I'll write ##A## for ##Tr(T)## and ##B## for ##Tr(T^2)##. Then subtract the RHS of the second from the RHS of the first and multiply the result by 2 to get:
    So twice the difference is zero.
    So the two RHSs are equal.
  4. Apr 12, 2016 #3
    Great! Thanks Andrew.
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