mM = (1/10) {{7, 11, 20, 
            17}, {-20, -40, -86, -74}, {0, -5, -10, -10}, {10, 28, 60, 53}}

MatrixForm[mM]

Eigensystem[mM]

evcs = Eigensystem[mM][[2]]

mP = 
      Transpose[{Re[evcs[[2]]], Im[evcs[[2]]], Re[evcs[[4]]], 
          Im[evcs[[4]]]}]

MatrixForm[Inverse[mP] . mM . mP]

Factor[Expand[Det[mM - x IdentityMatrix[4]]]]