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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:
    upload_2016-4-11_9-14-51.png
     
  2. jcsd
  3. Apr 11, 2016 #2

    andrewkirk

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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:
    $$2II_TA-2I_TB-A^3+3AB=2II_TA-2AB-A^3+3AB=A\left(2II_T-2B-A^2+3B\right)=A\left(2II_T+B-A^2\right)=2A\left(II_T-0.5(A^2-B)\right)
    =2A\left(II_T-II_T\right)=0$$
    So twice the difference is zero.
    So the two RHSs are equal.
     
  4. Apr 12, 2016 #3
    Great! Thanks Andrew.
     
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