Camera location	52° 44′ 24.84″ N, 8° 11′ 40.65″ E	View this and other nearby images on: OpenStreetMap	52.740233; 8.194626

m = RandomReal[{-10, 10}, {3, 3}];
m = Round[(m + Transpose[m])/2, 0.01];
\[Lambda] = Eigenvalues[m];
\[Psi] = Eigenvectors[m];

angle[u_, v_] := ArcCos[u.v/(Norm[u] Norm[v])];
xyz = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
\[Alpha]1 = \[Alpha] /. NMinimize[Norm[EulerMatrix[{\[Alpha], 0, 0}].xyz[[2]] - Cross[{0, 0, 1}, \[Psi][[3]]]/Norm[Cross[{0, 0, 1}, \[Psi][[3]]]] ], \[Alpha]][[2]];(*rotate y in the plane perpendicular to Subscript[\[Psi], z]*)
\[Beta]1 = angle[{0, 0, 1}, \[Psi][[3]]];
\[Gamma]1 = -angle[Cross[{0, 0, 1}, \[Psi][[3]]], \[Psi][[2]]];

p1 = Table[
   Grid[{{
      PaddedForm[MatrixForm[Inverse[EulerMatrix[{\[Alpha], 0, 0}]].m.EulerMatrix[{\[Alpha], 0, 0}] ], {3, 2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}], Sphere[{0, 0, 0}, 0.1]
        , Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha], 0, 0}].xyz[[j]]}], {j, 1, 3}] }, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
      }}]
   , {\[Alpha], 0, \[Alpha]1, \[Alpha]1/40}];
p2 = Table[
   Grid[{{
      PaddedForm[MatrixForm[Round[Inverse[EulerMatrix[{\[Alpha]1, \[Beta], 0}]].m.EulerMatrix[{\[Alpha]1, \[Beta], 0}], 0.01] ], {3,2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}], 
        Sphere[{0, 0, 0}, 0.1], Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha]1, \[Beta], 0}].xyz[[j]]}], {j, 1, 3}]
        }, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
      }}]
   , {\[Beta], 0, \[Beta]1, \[Beta]1/40}];
p3 = Table[
   Grid[{{
      PaddedForm[MatrixForm[Round[Inverse[EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}]].m.EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}], 0.01] ], {3, 2}], Graphics3D[{Gray, Table[Arrow[{{0, 0, 0}, xyz[[j]]}], {j, 1, 3}], Sphere[{0, 0, 0}, 0.1], Black, Table[Arrow[{{0, 0, 0}, EulerMatrix[{\[Alpha]1, \[Beta]1, \[Gamma]}].xyz[[j]]}], {j, 1, 3}]
        }, Boxed -> False, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
      }}]
   , {\[Gamma], 0, \[Gamma]1, \[Gamma]1/40}];
ListAnimate[Join[p1, p2, p3, Table[p3[[-1]], 30]]]

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