Can anyone suggest if the analytical expressions of the eigenvalues for this symmetry real matrix $L$ exist or not? All variables are real.
$$\begin{align} g_{11}&= a_{11}p^2+a_{66}q^2+a_{55}r^2+2.0a_{16}pq + 2.0a_{15}pr + 2.0a_{56}qr\\ g_{22}&= a_{66}p^2+a_{22}q^2+ a_{44}r^2+2.0a_{26}pq + 2.0a_{46}pr + 2.0a_{24}qr\\ g_{33}&= a_{55}p^2+a_{44}q^2+a_{33}r^2+2.0a_{45}pq + 2.0a_{35}pr + 2.0a_{34}qr\\ g_{12}&= a_{16}p^2+a_{26}q^2+a_{45}r^2+(a_{12}+a_{66})pq+(a_{14}+a_{56})pr+(a_{46} + a_{25})qr\\ g_{13} &= a_{15}p^2+a_{46}q^2+a_{35}r^2+(a_{14}+ a_{56})pq+(a_{13}+a_{55})pr + (a_{36} + a_{45})qr\\ g_{23} &=a_{56}p^2+a_{24}q^2+a_{34}r^2+(a_{46}+a_{25})pq+(a_{36} + a_{45})pr + (a_{23} + a_{44})qr\end{align}$$
$$L = \begin{bmatrix}g_{11}&g_{12}& g_{13}\\g_{12}& g_{22}& g_{23}\\g_{13}&g_{23}& g_{33}\end{bmatrix}$$
I've tried sympy, sage, and mathematica. None gave the results. So does it mean the analytical expressions don't exist? But if I substitute constants into $a_{ij}$ ($a_{11}$, $a_{12}$, etc.), the eigenvalue expressions can be found. Then why I can't have the expressions in terms of $a_{ij}$? Thanks very much.