and if M is of the form given above, you would need z_{2} = Az_{1},z_{3} = Az_{2} etc. This yields just such a cyclic
basis with the last entry on the left A^{kd}z_{1} which will then be equal to the appropriate linear combination of
lower powers times z_{1}.
If the blocks corresponding to ker
mi
(ϕi (A ) )
are ordered to decrease in size from upper left to lower
right, the matrix obtained is uniquely determined. This is shown later. This matrix is called the rational
canonical form.
Theorem 7.2.1Let A ∈ℒ
(V,V)
where V is a finite dimensional vector space and let its minimumpolynomial be
∏q mi
ϕi(λ)
i=1
Then there is a block diagonal matrix
( )
M1 0
|| .. ||
( . )
0 Mq
where M_{i}is the matrix of A with respect to ker
(ϕi(λ)mi)
taken with respect to a cyclic basis of the form
{βx1,⋅⋅⋅,βxp}
i
and it is of the form
( )
C1 0
|| ... ||
( )
0 Cmi
where each C_{j}is a companion matrix as described in ∗. If we arrange these C_{j}to be descending in sizefrom upper left to lower right, then the matrix just described is uniquely determined. (This proved later.)Also, the largest block corresponding to C_{k}is of size m_{k}d × m_{k}d.
Proof: The proof of uniqueness is given later. As to the last claim, let the cycles corresponding to
ker
mk
(ϕ (A ) )
be β_{y1},
⋅⋅⋅
,β_{yp}. If none of these cycles has length dm_{k}, then in fact, ker
mk
(ϕ (A) )
equals
ker
( )
ϕ(A)l
for some l< m_{k} which would contradict the fact shown earlier that the minimum polynomial
of A on ker
(ϕ (A )mk )
is ϕ
(λ)
^{mk}. ■
Note that there are exactly two things which determine the rational canonical form, the factorization of
the minimum polynomial into irreducible factors and the numbers consisting of
|βx|
for β_{x} a cycle in a
cyclic basis of V . Thus, if you can find these two things, you can obtain the rational canonical
form.
The important thing about this canonical form is that it does not depend on being able to factor the
minimum polynomial into linear factors. However, in most cases of interest when the field of
scalars is the complex numbers, a factorization of the minimum polynomial exists. When this
happens, there is a much more commonly used canonical form called the Jordan canonical
form.