Page 197 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
186
Now (ii) and 8.41 show that the last list above is a basis of null N, com-
pleting the proof of (b).
Suppose T ∈L(V). A basis of V is called a Jordan basis for T if
with respect to this basis T has a block diagonal matrix
 
A 1 0
 . 
 . .  ,
 
0 A m
where each A j is an upper-triangular matrix of the form
 
λ j 1 0
 . . 
 . . . . 
 
A j =   .
 . . 
 . 
 1 
0 λ j
To understand why In each A j , the diagonal is filled with some eigenvalue λ j of T, the line
each λ j must be an directly above the diagonal is filled with 1’s, and all other entries are 0
eigenvalue of T, (A j may be just a 1-by-1 block consisting of just some eigenvalue).
see 5.18. Because there exist operators on real vector spaces that have no
eigenvalues, there exist operators on real vector spaces for which there
is no corresponding Jordan basis. Thus the hypothesis that V is a com-
plex vector space is required for the next result, even though the pre-
vious lemma holds on both real and complex vector spaces.
The French 8.47 Theorem: Suppose V is a complex vector space. If T ∈L(V),
mathematician Camille then there is a basis of V that is a Jordan basis for T.
Jordan first published a
proof of this theorem Proof: First consider a nilpotent operator N ∈L(V) and the vec-
in 1870. tors v 1 ,...,v k ∈ V given by 8.40. For each j, note that N sends the first
vector in the list (N m(v j ) v j ,...,Nv j ,v j ) to 0 and that N sends each vec-
tor in this list other than the first vector to the previous vector. In other
words, if we reverse the order of the basis given by 8.40(a), then we ob-
tain a basis of V with respect to which N has a block diagonal matrix,
where each matrix on the diagonal has the form
 
0 1 0
 . . 
 . . 
 . . 
  .
 . 
 . . 
 1 
0 0