Computation of eigenvalues/vectors of a $9\times 9$ matrix

Question

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 $$ B := \begin{pmatrix} 0 & \nu_1 & \nu_2 & \nu_3 & 0 & 0 & \nu_3 & \nu_2 & \nu_1 \\ \nu_1 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \nu_2 \\ \nu_2 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \nu_3 \\ \nu_3 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ \nu_3 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ \nu_2 & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 \\ \nu_1 & \nu_2 & \nu_3 & 0 & 0 & 0 &0 & 0 & 0 \\ \end{pmatrix} $$

Answer 1

I hope someone can provide a better answer than this, but here's a start:

The standard way to solve linear algebra problems is using a library called LAPACK (http://www.netlib.org/lapack/)
There's a version of the LAPACK routines specifically designed for parallel processing across multiple computers - ScaLAPACK (http://www.netlib.org/scalapack/).
There's a description of how you might use something like ScaLAPACK on Amazon EC2 here: http://www.slideshare.net/pskomo... (it's not terribly satisfactory, but it's the best I could find)

There are specific routines available for various types of matrices (symmetric, hermitian, sparse, etc.) that run much faster than general routines, so the more you know about your matrix, the better. Hope it helps.

Answer 2

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.