 
Summary: Jordan Normal Form
§1. Jordan's Theorem
Definition The n by n matrix J,n with 's on the diagonal, 1's on the superdiagonal and 0's elsewhere is
called a Jordan block matrix. A Jordan matrix or matrix in Jordan normal form is a block matrix that is
has Jordan blocks down its block diagonal and is zero elsewhere.
Theorem Every matrix over C is similar to a matrix in Jordan normal form, that is, for every A there is
a P with J = P1
AP in Jordan normal form.
§2. Motivation for proof of Jordan's Theorem
Consider Jordan block A = J,n, for example,
A = J5,3 =
5 1 0
0 5 1
0 0 5
.
We see that
Ae1 = 5e1
