 Solution

A real symmetric traceless $2$ by $2$ matrix has eigenvalues $\lambda$ and $-\lambda$ and can be writtenfor a unit nematic director $\mathbf{n}\sim -\mathbf{n}$. Since $Q_{ij}{Q}_{ji}=2{\lambda }^2$, the uniform Landau free energy isFor $a<0<b$, its nonzero minima satisfyand henceThe signs $\pm {\lambda }_0$ amount to exchanging the two orthogonal eigenvectors, so the chosen convention takes ${\lambda }_0\ge 0$.