Problem in displaying output when finding eigenvalues of a Matrix

Question

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.

Answer 1

(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

X = Eigenvalues[M];

X[[1]]

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] &

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.