I am reading a book on linear algebra, and I found the following statement:

" Assume z is a simple eigenvalue of the matrix A. Because z is a simple eigenvalue, the core-nilpotent decomposition insures that A-zI is similar to a matrix of the form:

C 0

( ). "

0 01x1

I have trouble seeing why this is the case. This statement is used in the book to prove that the intersection between Range of (A-zI) and Nullspaceof (A-zI) is empty (so index(A-zI) = 1 and y*x <> 0 (where y is a left-hand eigenvector and x is an eigenvector of A). So I can't use these facts as assumptions to construct my argument!

Many thanks for your help!!!

Damien