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The characteristic polynomial is $(1/2-\lambda )(-1/2-\lambda )-2={\lambda }^2-9/4$. Its two roots give the eigenvalues

Multiply numerator and denominator by the complex conjugate of $1-i$:Thus the real part and imaginary part are respectively

The denominator $D=x^2+xy+y^2=(x+y/2{)}^2+3y^2/4$ is positive away from the origin. The partial derivatives areThey both vanish exactly onThese are lines of stationary points, with values $1/3$ and $-1$, respectively.