= Rotational invariant of a symmetric traceless tensor
{title2=$\operatorname{Tr}Q^k$}
= Rotational invariants of a symmetric traceless tensor
{synonym}
Under rotation by a <rotation matrix>, a <symmetric second-rank tensor> transforms by conjugation, so its traces of powers are invariant. For a three-dimensional <traceless second-rank tensor>, the <Cayley-Hamilton theorem> implies $\operatorname{Tr}Q^4=(\operatorname{Tr}Q^2)^2/2$. In two dimensions its <eigenvalues> are $s,-s$, so $\operatorname{Tr}Q^3=0$. These identities organize the <Landau-de Gennes free energy>.
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