Symbolic Math in a Computer Algebra System

Question

I am currently conducting my research on the spectrum of dephased of complex Hadamard matrices. Analytically, I was able to prove that two matrices A and B have equal non-zero trace and are equivalent to Tao’s isolated matrix $S_{6}$, then they share the same spectra. Now I would like to present the explicit calculation of the eigrnvalues for each possible trace. I tried doing this in MATLAB, and it gave me float-point numbers. I tried to use symbolic math, but I’m still unable to get the result the wanted.

An example is the following: Suppose I wanted to calculate for the eigenvalues of the matrix $$M:\begin{bmatrix} a & 0 \\ 0 & 0 \\ \end{bmatrix}$$ We can easily get that the eigenvalues are {a,0}.

The matrix $S_{6}$ consists of 1s and $\omega$s, where $\omega=e^{2 \pi i/6}$. Now I want to calculate for the eigenvalues of $S_{6}$, and I want it to be expressed in terms of $\omega$(for some) for an accurate result. Thanks.

Answer 1

Have you tried something like this?

function [M,V,D] = S6
  w = exp(2i*sym(pi)/3);
  M = [
    1 1 1 1 1 1;
    1 1 w w w^2 w^2;
    1 w 1 w^2 w^2 w;
    1 w w^2 1 w w^2;
    1 w^2 w^2 w 1 w;
    1 w^2 w w^2 w 1
    ];
  [V,D] = eig(M);
end