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Basically I am trying to find symbolic eigenvalues of a 4x4 Matrix. Using the below code

    {P, Q, 0, 0},
    {R, S, Subscript[θ, 1], Subscript[θ, 2]},
    {0, 0 , P - λ, Q},
    {σ Subscript[θ, 1], σ Subscript[θ, 2],
      R, S - λ}
   } );
X = Eigenvalues[M]

I expect to get the answer in a formated form, but instead I am getting the expression.

{Root[Q^2 R^2 - 2 P Q R S + P^2 S^2 + P Q R λ - 
    P^2 S λ + Q R S λ - P S^2 λ - 
    Q R λ^2 + 
    P S λ^2 + (-2 P - 2 S + 2 λ) #1^3 + #1^4 - 
    Q^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(1\), \(2\)]\) + 
    2 P Q σ Subscript[θ, 1] Subscript[θ, 2] - 
    Q λ σ Subscript[θ, 1] Subscript[θ, 
     2] - P^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + 
    P λ σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + #1^2 (P^2 - 
       2 Q R + 4 P S + S^2 - 3 P λ - 
       3 S λ + λ^2 - σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) + #1 (2 P Q R \
- 2 P^2 S + 2 Q R S - 2 P S^2 + P^2 λ - 2 Q R λ + 
       4 P S λ + S^2 λ - P λ^2 - 
       S λ^2 - 
       2 Q σ Subscript[θ, 1] Subscript[θ, 2] + 
       2 P σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) - λ \
σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) &, 1], 
 Root[Q^2 R^2 - 2 P Q R S + P^2 S^2 + P Q R λ - 
    P^2 S λ + Q R S λ - P S^2 λ - 
    Q R λ^2 + 
    P S λ^2 + (-2 P - 2 S + 2 λ) #1^3 + #1^4 - 
    Q^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(1\), \(2\)]\) + 
    2 P Q σ Subscript[θ, 1] Subscript[θ, 2] - 
    Q λ σ Subscript[θ, 1] Subscript[θ, 
     2] - P^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + 
    P λ σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + #1^2 (P^2 - 
       2 Q R + 4 P S + S^2 - 3 P λ - 
       3 S λ + λ^2 - σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) + #1 (2 P Q R \
- 2 P^2 S + 2 Q R S - 2 P S^2 + P^2 λ - 2 Q R λ + 
       4 P S λ + S^2 λ - P λ^2 - 
       S λ^2 - 
       2 Q σ Subscript[θ, 1] Subscript[θ, 2] + 
       2 P σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) - λ \
σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) &, 2], 
 Root[Q^2 R^2 - 2 P Q R S + P^2 S^2 + P Q R λ - 
    P^2 S λ + Q R S λ - P S^2 λ - 
    Q R λ^2 + 
    P S λ^2 + (-2 P - 2 S + 2 λ) #1^3 + #1^4 - 
    Q^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(1\), \(2\)]\) + 
    2 P Q σ Subscript[θ, 1] Subscript[θ, 2] - 
    Q λ σ Subscript[θ, 1] Subscript[θ, 
     2] - P^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + 
    P λ σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + #1^2 (P^2 - 
       2 Q R + 4 P S + S^2 - 3 P λ - 
       3 S λ + λ^2 - σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) + #1 (2 P Q R \
- 2 P^2 S + 2 Q R S - 2 P S^2 + P^2 λ - 2 Q R λ + 
       4 P S λ + S^2 λ - P λ^2 - 
       S λ^2 - 
       2 Q σ Subscript[θ, 1] Subscript[θ, 2] + 
       2 P σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) - λ \
σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) &, 3], 
 Root[Q^2 R^2 - 2 P Q R S + P^2 S^2 + P Q R λ - 
    P^2 S λ + Q R S λ - P S^2 λ - 
    Q R λ^2 + 
    P S λ^2 + (-2 P - 2 S + 2 λ) #1^3 + #1^4 - 
    Q^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(1\), \(2\)]\) + 
    2 P Q σ Subscript[θ, 1] Subscript[θ, 2] - 
    Q λ σ Subscript[θ, 1] Subscript[θ, 
     2] - P^2 σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + 
    P λ σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) + #1^2 (P^2 - 
       2 Q R + 4 P S + S^2 - 3 P λ - 
       3 S λ + λ^2 - σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) + #1 (2 P Q R \
- 2 P^2 S + 2 Q R S - 2 P S^2 + P^2 λ - 2 Q R λ + 
       4 P S λ + S^2 λ - P λ^2 - 
       S λ^2 - 
       2 Q σ Subscript[θ, 1] Subscript[θ, 2] + 
       2 P σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\) - λ \
σ 
\!\(\*SubsuperscriptBox[\(θ\), \(2\), \(2\)]\)) &, 4]}

However, if I assign numerical values to all the constants used and then use the command ClearAll["Global`*"];

and repeat the cell execution, I am getting the expected result (Formatted Math) which in Input form corresponds to

{1/2 (P + S - Sqrt[P^2 + 4 Q R - 2 P S + S^2]), 
 1/2 (P + S + Sqrt[P^2 + 4 Q R - 2 P S + S^2]), 
 1/2 (P + S - Sqrt[P^2 + 4 Q R - 2 P S + S^2] - 2 λ), 
 1/2 (P + S + Sqrt[P^2 + 4 Q R - 2 P S + S^2] - 2 λ)}

Which is what I wanted. I do not understand why assigning numerical values and then clearing them seems to solve the problem. Is there any way to avoid the problem in the first place? Note - The same problem does not occur when I take an arbitrary symbolic 2x2 matrix.

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5
  • $\begingroup$ Do you expect the eigenvalues to be independent of theta1 and theta2 (which seems unlikely). Is it possible you set theta1 and theta2 to zero, which simplifies the eigenvalues? The reason I ask is because ClearAll does not remove a definition like Subscript[[Theta], 1]=0. Instead, select Evaluation->Quit Kernel->Local to clear. $\endgroup$
    – bill s
    Jan 24, 2023 at 18:52
  • $\begingroup$ No, the eigenvalues should be dependent on theta_1 and theta_2, I understand that the simplification occurs only because they are taken as zero now. However my question still stands, how to get the output in a formated form as I get for smaller Matrices? $\endgroup$
    – Anik
    Jan 24, 2023 at 18:59
  • $\begingroup$ Why do you think there is a simpler form? It's the roots of a 4th order polynomial. $\endgroup$
    – bill s
    Jan 24, 2023 at 19:00
  • $\begingroup$ I understand the problem, I was expecting a closed answer that did not involve the Root[] function. Apparently, that is not possible. Thanks !! $\endgroup$
    – Anik
    Jan 24, 2023 at 19:06
  • $\begingroup$ It might be possible to represent things in a different way... the first two answers to this question are pretty informative about working with root objects... mathematica.stackexchange.com/a/126156/1783 $\endgroup$
    – bill s
    Jan 24, 2023 at 20:10

1 Answer 1

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(M = {{P, Q, 0, 0}, 
      {R, S, Subscript[θ, 1], Subscript[θ, 2]}, 
      {0, 0, P - λ, Q}, 
     {σ Subscript[θ, 1], σ Subscript[θ, 2], R, S - λ}}) //
 MatrixForm

enter image description here

X = Eigenvalues[M];

X[[1]]

enter image description here

The Root expressions are a compact representation. They can be converted to radical representation by ToRadicals

X // LeafCount

(* 1033 *)

(X2 = X // ToRadicals) // LeafCount

(* 33817 *)

X2[[1]] // Short[#, 6] &

enter image description here

The expanded form of the expressions is too large to be of practical use. The Root expressions convey the same info, i.e., complicated expression.

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