Matrix Eigenvalues Eigenvectors

Initialization Steps

mtemp = DiagonalMatrix[{2 Pi, 4}];
mtemp // MatrixForm

<< "Geometry`Rotations`"

MatrixForm[
      msim = Transpose[RotationMatrix2D[Pi/4]] . mtemp . 
          RotationMatrix2D[Pi/4]]

mymatrix  =  {{2 + Pi, -2 + Pi}, {-2 + Pi, 
            2 + Pi}};
mymatrix // MatrixForm

Solve[
      Det[mymatrix  -  λ IdentityMatrix[2]] ==  
        0, λ]

Eigenvectors[mymatrix]

{evec1,  evec2}  =  Eigenvectors[mymatrix]

Eigensystem[mymatrix]

shows that the eigenvectors corresponding  2 π is (1,1)

simtrans =  {evec2, evec1} // 
          Transpose;
simtrans // MatrixForm

Inverse[simtrans] . mymatrix . simtrans // Simplify // 
      MatrixForm

RotationMatrix2D[-Pi/4] // MatrixForm

Shows that the transformation to the diagonal basis is a rotation of -π/4

Which makes sense considering in initialization steps that mymatrix was created with a rotation on π/4 of a diagonal matrix

<< LinearAlgebra`Orthogonalization`

GramSchmidt[Eigenvectors[mymatrix]]

evec1
evec2

mymatrix . evec1

mymatrix . evec2

MatrixPower[mymatrix, 12] . evec2 // Simplify