As Bob Hanlon comments the original output without Simplify
is already in the form you want:
efre = Sqrt[-coe // Eigenvalues]
{Sqrt[1/2 (5 + Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (3 + Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (5 - Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (3 - Sqrt[5])] Sqrt[k/m]}
However, it may help to understand how to work with Simplify
and FullSimplify
.
As elsewhere(1),(2) you need to specify a ComplexityFunction
that is closer to the way you see expressions rather than Mathematica's internal one, which is roughly approximated by LeafCount
.
For the sake of an example let's transform our output into a different form so that we can see that Simplify
has an effect:
e2 = ExpandAll //@ efre
{Sqrt[5/2 + Sqrt[5]/2] Sqrt[k/m],
Sqrt[3/2 + Sqrt[5]/2] Sqrt[k/m],
Sqrt[(5 k)/m - (Sqrt[5] k)/m]/Sqrt[2],
Sqrt[(3 k)/m - (Sqrt[5] k)/m]/Sqrt[2]}
Now we Simplify
with a ComplexityFunction
that is based on visual output size:
Simplify[e2, ComplexityFunction -> Composition[StringLength, ToString]]
{Sqrt[1/2 (5 + Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (3 + Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (5 - Sqrt[5])] Sqrt[k/m],
Sqrt[1/2 (3 - Sqrt[5])] Sqrt[k/m]}
// Simplify
with//N// Simplify
and they look the same. $\endgroup$