Example 3

  Clear[a,S]

  a={{2,-4,2,2},{-2,0,1,3},{-2,-2,3,3},{-2,-6,3,7}};
  MatrixForm[a]

  Eigenvalues[a]

You notice that a has a repeated eigenvalue. This does not necessarily mean that a does not have a full complement of eigenvectors. Let's see.

  Eigenvectors[a]

Notice that the eigenvalue 2 has a full complement of eigenvectors, but the eigenvalue 4 does not.

  LinearSolve[a-4 IdentityMatrix[4],{0,1,1,1}]

  S=Transpose[{{0,1,1,1},{1,-1,-1,0},{2,1,0,2},{0,1,2,0}}];
  MatrixForm[S]

  Ja=Inverse[S].a.S;
  MatrixForm[Ja]

The following finds the characteristic polynomial of a.

  Det[a-x IdentityMatrix[4]]

  Factor[Det[a-x IdentityMatrix[4]]]

Some shortcuts.

  am2I=a-2IdentityMatrix[4];
  am4I=a-4IdentityMatrix[4];

Let's check (-4+x)(-2+x).

  MatrixForm[am4I.am2I]

Let's check (-4+x)^2 (-2+x).

  MatrixForm[am4I.am4I.am2I]

Bingo, the minimal polynomial is (suitably written to be monic):
(x-4)^2 (x-2)

Recall the Jordan form of a.

  MatrixForm[Ja]

Compare the size of the Jordan block for each eigenvalue with the multiplicity of that eigenvalue in the MINIMAL polynomial.

Up to Jordan Form and Minimal Polynomials