mM = Table[1, {j, 1, 11}, {k, 1, 11}]

Det[mM - x IdentityMatrix[11]]

mM

Eigensystem[mM]

mP = {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, -1, 0, 0, 0, 0, 0, 0, 
          0, 0, 0}, {1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, -1, 
          0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0}, {1, 
          0, 0, 0, 0, -1, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, -1, 0, 
          0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0}, {1, 0, 0, 0, 0, 
          0, 0, 0, -1, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0}, {1, 
          0, 0, 0, 0, 0, 0, 0, 0, 0, -1}}

Inverse[mP] . mM . mP

11 Inverse[mP]

mP1 = 
      Transpose[{{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {-1, 0, 0, 0, 0, 0, 
            0, 0, 0, 0, 1}, {-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {-1, 
            0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {-1, 0, 0, 0, 0, 0, 0, 1, 0, 
            0, 0}, {-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {-1, 0, 0, 0, 
            0, 1, 0, 0, 0, 0, 0}, {-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 
            0}, {-1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 1, 0, 0, 
            0, 0, 0, 0, 0, 0}, {-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
            0}}]

MatrixForm[Inverse[mP1] . mM . mP1]

MatrixForm[{{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 
          7}}]

Eigensystem[{{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 
          7}}]

mP2 = 
      Transpose[{{-(19/14) + (3/28) (8 + 2 (Sqrt[21])), -(4/7) + 
              (1/14) (8 + 2 (Sqrt[21])), (3/14) + (1/28) (8 + 2 (Sqrt[21])), 
            1}, {-(19/14) + (3/28) (8 - 2 (Sqrt[21])), -(4/7) + 
              (1/14) (8 - 2 (Sqrt[21])), (3/14) + (1/28) (8 - 2 (Sqrt[21])), 
            1}, {2, -3, 0, 1}, {1, -2, 1, 0}}]

(Inverse[
          mP2] . {{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}} . 
        mP2)

Chop[
      N[(Inverse[
              mP2] . {{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 
                7}} . mP2) ]]

Simplify[(Inverse[
            mP2] . {{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}} . 
          mP2) ]

11^10

MatrixForm[MatrixPower[mM, 11]]

MatrixForm[MatrixPower[mM, 2]]