Basics of the Jordan Canonical Form

  
  b=MatrixForm[{{2,1,0,0,0},{0,2,1,0,0},{0,0,2,0,0},{0,0,0,3,1},{0,0,0,0,3}}]
  

This 5 by 5 matrix consists of two Jordan blocks, one (3 by 3) corresponding to the eigenvalue 2 and the other (2 by 2) corresponding to the eigenvalue 3.

To be more precise, a matrix in Jordan form has the following properties :

a. Every entry not on the diagonal or superdiagonal is zero. The entries on the diagonal are the eigenvalues, repeated according to their multiplicity in the characteristic polynomial.

b. There is one Jordan block corresponding to each eigenvector.

c. Each eigenvalue appears along the main diagonal the number of times of it's multiplicity in the characteristic polynomial.

d. The size of the Jordan block is related to the minimal polynomial, as we shall see later.