Error in computation of eigenvalues in Sage

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Hello,

It seems that Sage is not able to solve your equation symbolically. However, you can compute the characteristic polynomial and try to analyze it

R = ZZ['x,sigma,gamma,omega,Q1,Q2,Q3,Q4,A1,A2,A3,mu_v']
R.inject_variables()
J = matrix(R,[[-Q1,0,0,0,omega,0,0,0,0,-A1],
  [0,-Q2,0,0,0,omega,0,0,0,A1],
  [0,gamma,Q3,0,0,0,omega,0,0,0],
  [0,0,sigma,-Q1,0,0,0,omega,0,0],
  [A2,0,0,0,-Q4,0,0,0,0,0],
  [0,A2,0,0,0,-Q4,0,0,0,0],
  [0,0,A2,0,0,0,-Q4,0,0,0],
  [0,0,0,A2,0,0,0,-Q4,0,0],
  [0,0,-A3,0,0,0,0,0,-mu_v,0],
  [0,0,A3,0,0,0,0,0,0,-mu_v]])
p = J.charpoly().subs(R('x'))
p.factor()

As you can notice I did not use symbolic variables but polynomial variables that are more powerful (and more predictable). In particular, the factorization with p.factor() takes place in Z[x,sigma,gamma,Q1,...]`.

The answer of the above code shows two small factors

x + mu_v
(-x^2 - x*Q1 - x*Q4 - Q1*Q4 + omega*A2)^2

and a big factor of degree 5. The small factors give you already three eigenvalues (five if counted with multiplicity). The big factor is quite complicated and I doubt there would be any formula.

Vincent

This seems to be a bug in maxima, which should be reported. A little googling shows this error has happened before but not with a very clear input matrix. This script in maxima leads straight to a (hopefully debuggable) error:

M: matrix([-d, 0, 0, 0, v, 0, 0, 0, 0, -a], [0, -e, 0, 0, 0, v, 0, 0, 0, a], [0, w, f, 0, 0, 0, v, 0, 0, 0], [0, 0, t, -d, 0, 0, 0, v, 0, 0], [b, 0, 0, 0, -g, 0, 0, 0, 0, 0], [0, b, 0, 0, 0, -g, 0, 0, 0, 0], [0, 0, b, 0, 0, 0, -g, 0, 0, 0], [0, 0, 0, b, 0, 0, 0, -g, 0, 0], [0, 0, -c, 0, 0, 0, 0, 0, -u, 0], [0, 0, c, 0, 0, 0, 0, 0, 0, -u]);
eigenvalues(M);

As a workaround, you can use:

sage: maxima_calculus(M.charpoly()).factor()

which shows you that the characteristic polynomial is of the form linearquadratic^2quintic

sage: M.charpoly().roots()
[(-1/2*Q1 - 1/2*Q4 - 1/2*sqrt(Q1^2 - 2*Q1*Q4 + Q4^2 + 4*A2*omega), 2),
 (-1/2*Q1 - 1/2*Q4 + 1/2*sqrt(Q1^2 - 2*Q1*Q4 + Q4^2 + 4*A2*omega), 2),
 (-mu_v, 1)]

gives you 3 eigenvalues. The eigenvalue corresponding to the quintic factor is not reported here, likely because maxima (which is used for this if I'm not mistaken) doesn't have access to the means to express that eigenvalue properly.