# Finding the (symbolic) eigenvalues of a matrix with some assumption

I have the following 3$$\times$$3 symmetric matrix with symbolic entries.

Meff = $$\begin{array}{ccc} 2 \beta \text{c12}^2 \text{c13}^2 \text{\eta 11} \text{m1}+\alpha \text{c13}^2 \text{D31} \text{s12}^2+2 \beta \text{c13}^2 \text{\eta 11} \text{m2} \text{s12}^2+\text{D31} \text{s13}^2+2 \beta \text{\eta 11} \text{m3} \text{s13}^2 & \alpha \text{c12} \text{c13} \text{D31} \text{s12}-\beta \text{c12} \text{c13} \text{\eta 11} \text{m1} \text{s12}+\beta \text{c12} \text{c13} \text{\eta 11} \text{m2} \text{s12} & \beta \text{c12}^2 (-\text{c13}) \text{\eta 11} \text{m1} \text{s13}-\alpha \text{c13} \text{D31} \text{s12}^2 \text{s13}+\text{c13} \text{D31} \text{s13}-\beta \text{c13} \text{\eta 11} \text{m2} \text{s12}^2 \text{s13}+\beta \text{c13} \text{\eta 11} \text{m3} \text{s13} \\ \alpha \text{c12} \text{c13} \text{D31} \text{s12}-\beta \text{c12} \text{c13} \text{\eta 11} \text{m1} \text{s12}+\beta \text{c12} \text{c13} \text{\eta 11} \text{m2} \text{s12} & \alpha \text{c12}^2 \text{D31} & \alpha (-\text{c12}) \text{D31} \text{s12} \text{s13} \\ \beta \text{c12}^2 (-\text{c13}) \text{\eta 11} \text{m1} \text{s13}-\alpha \text{c13} \text{D31} \text{s12}^2 \text{s13}+\text{c13} \text{D31} \text{s13}-\beta \text{c13} \text{\eta 11} \text{m2} \text{s12}^2 \text{s13}+\beta \text{c13} \text{\eta 11} \text{m3} \text{s13} & \alpha (-\text{c12}) \text{D31} \text{s12} \text{s13} & \text{c13}^2 \text{D31}+\alpha \text{D31} \text{s12}^2 \text{s13}^2 \\ \end{array}$$

I want the eigenvalues of the above matrix but Mathematica gives very long expressions for eigenvalues if I use Eigenvalues[Meff, Cubics -> True] which is not understandable. I want to make certain approximations, for example, I want to ignore some terms in eigenvalues that are in higher powers of say $$\alpha$$ and $$s_{13}$$ and so on just like we truncate the expressions up to certain orders in some parameters. How to do that? Sorry if I am not able to make the problem understandable. Below is the Mathematica code for the matrix (Meff)

R13 = {{c13, 0, Exp[-I*z13]*s13}, {0, 1, 0}, {-Exp[I*z13]*s13, 0,
c13}};
U13 = {{c13, 0, s13}, {0, 1, 0}, {-s13, 0, c13}};
R12 = {{c12, s12, 0}, {-s12, c12, 0}, {0, 0, 1}};
R23 = {{1, 0, 0}, {0, c23, s23}, {0, -s23, c23}} ;
Dnu = {{m1, 0, 0}, {0, m2, 0}, {0, 0, m3}};
Mass = {{0, 0, 0}, {0, D31*α, 0}, {0, 0, D31}};
Udelta = {{1, 0, 0}, {0, 1, 0}, {0, 0, Exp[I*z13]}} ;
Udeltadag = {{1, 0, 0}, {0, 1, 0}, {0, 0, Exp[-I*z13]}} ;
UBdag = Assuming[{s12, s12, s23, c12, c13, c23, z13, α,
D31, β, m1, m2,
m3, η11, η22, η33} ∈ Reals,
TrigToExp[Simplify@ComplexExpand[ConjugateTranspose[UB]]]];
delm1 = β*{{η11, 0, 0}, {0, 0, 0}, {0, 0, 0}};
M1 = UB.Dnu.UBdag.delm1 + delm1.UB.Dnu.UBdag;
Mtotal =  UB.Mass.UBdag + M1 ;
Meff =  Assuming[{s12, s12, s23, c12, c13, c23, z13, α,
D31, β, m1, m2,
m3, η11, η22, η33} ∈ Reals,
TrigToExp[FullSimplify@ComplexExpand[Mtotal]]]
Eigenvalues[Meff, Cubics -> True]

• Your code doesn't work and Eigenvalues stays unevaluated. You need to remove // MatrixForm from Meff - it's for presentation purposes only. Dec 28, 2022 at 14:27
• Done. Can you try with "Mtotal" it is same as "Meff"? Dec 28, 2022 at 14:34
• You can start off with something like this eigs /. {Power[s12, n_] /; n > 1 :> 0, Power[\[Alpha], n_] /; n > 1 :> 0} which will work in most cases except for where you have radicals. Dec 28, 2022 at 14:35
• Maybe use Series on the eigenvalues Dec 28, 2022 at 15:10
• @flinty Thank you for the suggestion. And yes, this approach seems to work a bit. Can you tell me if I want to ignore some combinations of terms lets us say I want to ignore products of $\alpha\times\beta\times m1$ and $\alpha \times \beta\times \eta$ and so on, then what command should I give? Dec 29, 2022 at 10:28