Simplify or Reduce or Replace the Eigenvalues of a Matrix

Question

I have a $3 \times 3$ matrix which I define like so :

F = {{ω^2 + 1/2 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) (1 + \
ϵ), -I ω Subscript[ω, c] Cos[θ], 
    I ω Subscript[ω, c]
      Sin[θ] Sin[ϕ]}, {I ω Subscript[ω, c]
      Cos[θ], ω^2 + 1/2 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) (1 - \
ϵ), -I ω Subscript[ω, c]
      Sin[θ] Cos[ϕ]}, {-I ω Subscript[ω, c]
      Sin[θ] Sin[ϕ], 
    I ω Subscript[ω, c]
      Sin[θ] Cos[ϕ], ω^2 - 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\)}};

And I want to determine the Eigenvalues of this matrix, I do this with:

Eigenvalues[F, Cubics -> True]

This works fine and I get the three eigenvalues, however if you run this code you will note that the eigenvalues are rather unpleasant. There are also some common factors that appear in all three of the equations.

I was wondering if there is a way I can either simplify these (using FullSimplify[...] does not work it just runs continuously) or use Replace[...] some how to just put the common factors into a constant and make the equations nicer this way.

Any ideas?

Answer 1

There is no special trick to force Mma to find the factors instead of you. However, something one can do in your case. These are two eigenvalues:

λ1 = Map[Simplify, Eigenvalues[F, Cubics -> True]][[1]]
λ2 = Map[Simplify, Eigenvalues[F, Cubics -> True]][[2]]

These are two replacement rules:

rule1 = 1/3 Sqrt[729 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(4\)]\) (ω^2 (2 \
+ 6 Cos[2 θ] + ϵ Cos[2 (θ - ϕ)] - 
           2 ϵ Cos[2 ϕ] + ϵ Cos[
             2 (θ + ϕ)]) 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) + 
        2 (-1 + ϵ^2) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(4\)]\))^2 - 
     108 (4 ω^2 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) + (3 + \
ϵ^2) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(4\)]\))^3] -> x;
rule2 = 18 ω^2 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) + 
    54 ω^2 Cos[2 θ] 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) + 
    9 ϵ ω^2 Cos[2 (θ - ϕ)] 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) - 
    18 ϵ ω^2 Cos[2 ϕ] 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) + 
    9 ϵ ω^2 Cos[2 (θ + ϕ)] 
\!\(\*SubsuperscriptBox[\(ω\), \(c\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(2\)]\) - 18 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(6\)]\) + 
    18 ϵ^2 
\!\(\*SubsuperscriptBox[\(ω\), \(z\), \(6\)]\) -> y;

Now

λ1 /. rule1 /. rule2

yields

while

λ2 /. rule1 /. rule2

gives

Hope this helps. Have fun!