Linear Algebra, Theory And Applications, 2012a

256 LINEAR TRANSFORMATIONS CANONICAL FORMS
By Theorem 10.2.6, the matrix of N with respect to the above basis is the block diagonal
matrix
⎛
⎞
M 1 0
⎜
⎝
. ..
⎟
⎠
0 M q
where M k denotes the matrix of N restricted to span ( )
β xk . In computing this matrix, I
will order β xk
as follows:
(
N
r k −1 )
x k , ··· ,x k
Also the cyclic sets β x1
,β x2
, ··· ,β xq
will be ordered according to length, the length of
β xi
being at least as large as the length of β xi+1
. Then since N r k
x k =0, it is now easy
to find M k . Using the procedure mentioned above for determining the matrix of a linear
transformation, ( ) 0 N
r k −1 x k ··· Nx k =
⎛
⎞
0 1 0
( )
.
N
r k −1 x k N rk−2 x k ··· x k 0 0 ..
⎜
⎝
. ⎟
. . .. 1 ⎠
0 0 ··· 0
Thus the matrix M k is the r k ×r k matrix which has ones down the super diagonal and zeros
elsewhere. The following convenient notation will be used.
Definition 10.4.3 J k (α) is a Jordan block if it is a k × k matrix of the form
⎛
⎞
α 1 0
.
J k (α) =
0 .. . ..
⎜
⎝
.
. .. . ⎟ .. 1 ⎠
0 ··· 0 α
In words, there is an unbroken string of ones down the super diagonal and the number α
filling every space on the main diagonal with zeros everywhere else.
Then with this definition and the above discussion, the following proposition has been
proved.
Proposition 10.4.4 Let N ∈L(W, W ) be nilpotent,
N m =0
for some m ∈ N. Here W is a p dimensional vector space with field of scalars F. Then there
exists a basis for W such that the matrix of N with respect to this basis is of the form
⎛
⎞
J r1 (0) 0
J r2 (0)
J = ⎜
⎝
. ..
⎟
⎠
0 J rs (0)
where r 1 ≥ r 2 ≥···≥r s ≥ 1 and ∑ s
i=1 r i = p. In the above, the J rj (0) is a Jordan block of
size r j × r j with 0 down the main diagonal.