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6.3. Similarity Invariants and Jordan Reduction
Definition 6.3.2 The elementary divisors of the matrix M ∈ M n (k) are
α(j,k)
for which the exponent α(j, k) is nonzero. The mul-
the polynomials p
k
m
is the number of solutions j of the
tiplicity of an elementary divisor p
k
equation α(j, k)= m.The list of elementary divisors is the sequence of
these polynomials, repeated with their multiplicities.
Let us begin with the case of the companion matrix N of some polynomial
P. Its similarity invariants are (1,... , 1,P)(see above). Let Q 1 ,... ,Q t be
its elementary divisors (we observe that each has multiplicity one). We then
have P = Q 1 ··· Q t, while the Q l ’s are pairwise coprime. To each Q l we
associate its companion matrix N l , and we form a block-diagonal matrix
N := diag(N 1 ,... ,N t ). Since each N l − XI l is equivalent to a diagonal

matrix
 
1
.
 . 
 . 
 
 1 
Q l
in M n(l) (k[X]), the whole matrix N − XI n is equivalent to

 
1
.
 . 
 . O 
 
1
 
Q :=   .
Q 1
 
 
.
 . 
 O . 
Q t

Let us now compute the similarity invariants of N , that is, the invariant
factors of Q. It will be enough to compute the greatest common divisor
D n−1 of the minors of size n − 1. Taking into account the principal minors
of Q,we see that D n−1 must divide every product of the form

Q l , 1 ≤ k ≤ t.
l=k

Since the Q l ’s are pairwise coprime, this implies that D n−1 =1. This

means that the list of similarity invariants of N has the form (1,... , 1, ·),
where the last polynomial must be the characteristic polynomial of N .

This polynomial is the product of the characteristic polynomials of the
N l ’s. These being equal to the Q l ’s, the characteristic polynomial of N is

P. Finally, N and N have the same similarity invariants and are therefore

similar.
Now let M be a general matrix in M n (k). We apply the former reduction
to every diagonal block M j of its Frobenius canonical form. Each M j is
similar to a block-diagonal matrix whose diagonal blocks are companion
matrices corresponding to the elementary divisors of M entering into the