where I denotes the identity linear transformation. Without loss of generality, let the
dimensions of the V_{k} be decreasing from left to right. These V_{k} are called the generalized
eigenspaces.
It follows from the definition of V_{k} that
(A − λkI)
is nilpotent on V_{k} and clearly each V_{k} is A
invariant. Therefore from Proposition 9.4.4, and letting A_{k} denote the restriction of A to V_{k}, there
exists an ordered basis for V_{k},β_{k} such that with respect to this basis, the matrix of
(Ak − λkI)
is
of the form given in that proposition, denoted here by J^{k}. What is the matrix of A_{k} with respect
to β_{k}? Letting
{b1,⋅⋅⋅,br}
= β_{k},
∑ k ∑ ∑ ( k )
Akbj = (Ak − λkI)bj +λkIbj ≡ Jsjbs + λkδsjbs = Jsj + λkδsj bs
s s s
and so the matrix of A_{k} with respect to this basis is J^{k} + λ_{k}I where I is the identity
matrix.
Therefore, with respect to the ordered basis
{β1,⋅⋅⋅,βr}
the matrix of A is in Jordan
canonical form. This means the matrix is of the form
This proves the existence part of the following fundamental theorem.
Note that if any of the β_{k} consists of eigenvectors, then the corresponding Jordan block will
consist of a diagonal matrix having λ_{k} down the main diagonal. This corresponds to m_{k} = 1. The
vectors which are in ker
(A − λkI)
^{mk} which are not in ker
(A − λkI)
are called generalized
eigenvectors.
The following is the main result on the Jordan canonical form.
Theorem 9.5.2Let V be an n dimensional vector space with field of scalars ℂ or someother field such that the minimal polynomial of A ∈ℒ
(V,V )
completely factors into powersof linear factors. Then there exists a uniqueJordan canonical form for A as described in9.6- 9.8, where uniqueness is in the sense that any two have the same number and size ofJordan blocks.
Proof: It only remains to verify uniqueness. Suppose there are two, J and J^{′}. Then these are
matrices of A with respect to possibly different bases and so they are similar. Therefore, they
have the same minimal polynomials and the generalized eigenspaces have the same
dimension. Thus the size of the matrices J
(λk)
and J^{′}
(λk)
defined by the dimension of these
generalized eigenspaces, also corresponding to the algebraic multiplicity of λ_{k}, must
be the same. Therefore, they comprise the same set of positive integers. Thus listing
the eigenvalues in the same order, corresponding blocks J
(λk)
,J^{′}
(λk)
are the same
size.
It remains to show that J
(λk)
and J^{′}
(λk)
are not just the same size but also are the same up
to order of the Jordan blocks running down their respective diagonals. It is only necessary to
worry about the number and size of the Jordan blocks making up J
(λk)
and J^{′}
(λk)
. Since J,J^{′}
are similar, so are J − λ_{k}I and J^{′}− λ_{k}I.
and it is required to verify that p = r and that the same blocks occur in both. Without loss of
generality, let the blocks be arranged according to size with the largest on upper left corner falling
to smallest in lower right. Now the desired conclusion follows from Corollary 9.4.6.
■
Note that if any of the generalized eigenspaces ker
(A − λ I)
k
^{mk} has a basis of eigenvectors,
then it would be possible to use this basis and obtain a diagonal matrix in the block
corresponding to λ_{k}. By uniqueness, this is the block corresponding to the eigenvalue λ_{k}.
Thus when this happens, the block in the Jordan canonical form corresponding to λ_{k} is
just the diagonal matrix having λ_{k} down the diagonal and there are no generalizedeigenvectors.
The Jordan canonical form is very significant when you try to understand powers of a matrix. There exists
an n×n matrix S^{1}
such that
A = S −1JS.
Therefore, A^{2} = S^{−1}JSS^{−1}JS = S^{−1}J^{2}S and continuing this way, it follows
Ak = S −1JkS.
where J is given in the above corollary. Consider J^{k}. By block multiplication,
( )
Jk1 0
Jk = || .. || .
( . )
0 Jkr
The matrix J_{s} is an m_{s}× m_{s} matrix which is of the form
Js = D + N
for D a multiple of the identity and N an upper triangular matrix with zeros down the main
diagonal. Thus N^{ms} = 0. Now since D is just a multiple of the identity, it follows that DN = ND.
Therefore, the usual binomial theorem may be applied and this yields the following equations for
k ≥ m_{s}.
k ( )
k k ∑ k k−j j
Js = (D + N ) = j=0 j D N
ms ( )
= ∑ k Dk− jN j, (9.9)
j=0 j
the third equation holding because N^{ms} = 0. Thus J_{s}^{k} is of the form