# I Third Invariant expressed with Cayley-Hamilton Theorem

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1. Apr 11, 2016

### FluidStu

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

with 1.269 being:

2. Apr 11, 2016

### andrewkirk

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.

3. Apr 12, 2016

### FluidStu

Great! Thanks Andrew.