TABLE OF CONTENTS The goal of this section is to do for general matrices A ? C n n what was done for nilpotent matrices in 7.7-reduce A by means of a similarity transformation to a block-diagonal matrix in which each block has a simple triangular form. The two major components for doing this are now in place-they are the corenilpotent decomposition (p. 397) and the Jordan form for nilpotent matrices. All that remains is to connect these two ideas. To do so, it is convenient to adopt the following terminology.
The index of an eigenvalue ? for a matrix A ? C n n is defined to be the index of the matrix ( A ? ? I). In other words, from the characterizations of index given on p. 395, index ( ?) is the smallest positive integer k such that any one of the following statements is true.
rank (( A ? ? I) k) = rank (( A ? ? I) k +1).
R (( A ? ? I) k) = R (( A ? ? I) k +1).
N (( A ? ? I) k) = N (( A ? ? I) k +1).
R (( A ? ? I) k) ? N (( A ? ? I
