The ellipse $x^2+xy+y^2=1$

Matrix. $A=\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix}$.

Eigenvalues. $\det(A-\lambda I)=(1-\lambda)^2-1/4=\lambda^2-2\lambda+3/4$. Roots: $\lambda=(2\pm1)/2$, so $\lambda_1=1/2$ and $\lambda_2=3/2$.

Diagonalization. In the orthonormal eigenbasis, $Q=\frac12 u^2+\frac32 v^2$. The equation becomes
$$\frac{u^2}{2}+\frac{v^2}{2/3}=1.$$

Geometry. This is an ellipse centered at the origin, with semi-axes $\sqrt2$ (in the direction of the eigenvector for $\lambda_1=1/2$) and $\sqrt{2/3}$ (in the direction for $\lambda_2=3/2$).

Conclusion. $E$ is an ellipse; its semi-axes are $\sqrt2$ and $\sqrt{2/3}$.