Computation of eigenvalues/vectors of a $9\times 9$ matrix I have a symmetric matrix (with real coefficients) and I need to compute its eigenvalues and eigenvectors. My matrix depends on 3 parameters $(\nu_1,\nu_2,\nu_3)$ that are not independent (in fact we have $\nu_1^2+\nu_2^2+\nu_3^2=1$). If I am using Maple to compute the eigenvalues/vectors, I get an ugly answer since I don't know how to use the fact that I know that $\nu_1^2+\nu_2^2+\nu_3^2=1$ and consequently, there is certainly a lot of simplification that Maple doesn't do.
Is there a software that I can use to do this kind of computation? 
For example, how could I if the matrix is
$$ 
B := \begin{pmatrix}
0 & \nu_1 & \nu_2 & \nu_3 & 0 & 0 & \nu_3 & \nu_2 & \nu_1 \\
\nu_1 & 0 & 0   & 0 & 0 & 0  &0 & 0 & \nu_2 \\
\nu_2 & 0 & 0   & 0 & 0 & 0  &0 & 0 & \nu_3 \\
\nu_3 & 0 & 0   & 0 & 0 & 0  &0 & 0 & 0 \\
0 & 0 & 0   & 0 & 0 & 0  &0 & 0 & 0 \\
0 & 0 & 0   & 0 & 0 & 0  &0 & 0 & 0 \\
\nu_3 & 0 & 0   & 0 & 0 & 0  &0 & 0 & 0 \\
\nu_2 & 0 & 0   & 0 & 0 & 0  &0 & 0 & 0 \\
\nu_1 & \nu_2 & \nu_3   & 0 & 0 & 0  &0 & 0 & 0 \\
\end{pmatrix}
$$

One can do this with most of the CASs, by taking the condition $a^2+b^2+c^2-1:=0$ as another polynomial equation for computing a Gröbner basis. It depends on your specific matrix, whether or not the complexity of the system of polynomial equations is still manageable.
Edit: the characteristic polynomial of $B$ is given by
$$
f(t)=t^9 - 2t^7(a^2 + b^2 + c^2) - 2t^6ab(a + c) + t^5(a^2c^2 - 2ab^2c +
b^4 + b^2c^2 + c^4).
$$
Using $a^2+b^2+c^2=1$ we obtain
$$
f(t):=t^5 \cdot(t^4 - 2t^2 + 2tb( - ac + b^2 + c^2 - 1) - 2ab^2c + b^4 + c^2).
$$
So we have $\lambda=0$ eigenvalue with multiplicity $5$, and the other four eigenvalues the zeroes of the polynomial in brackets.
