Page 124 - Matrices theory and applications

6.3. Similarity Invariants and Jordan Reduction
(j)
by M
(k) the companion matrix of the polynomial P j .Let us
∈ M n j
form the matrix M , block-diagonal, whose diagonal blocks are the M
’s.

The few first polynomials P j are generally constant (we shall see below
that the only case where P 1 is not constant corresponds to M = αI n ), and
the corresponding blocks are empty, as are the corresponding rows and
columns. To be precise, the actual number m of diagonal blocks is equal to
the nuber of nonconstant similarity invariants. (j) 107
− M (j) is equivalent to the matrix N (j) =
Since the matrix XI n j
diag(1,... , 1,P j ), we have
− M (j) = P (j) N (j) Q (j) ,
XI n j
where P (j) ,Q (j) ∈ GL n j (k[X]). Let us form matrices P, Q ∈ GL n (k[X])
by
P = diag(P (1) ,... ,P (n) ), Q =diag(Q (1) ,... ,Q (n) ).
We obtain
XI n − M = PNQ, N = diag(N (1) ,... ,N (n) ).

Here N is a diagonal matrix, whose diagonal entries are the similarity
invariants of M, up to the order. In fact, each nonconstant P j appears
in the associated block N (j) . The other diagonal terms are the constant
1, which occurs n − m times; these are the polynomials P 1 ,... ,P n−m ,as
expected. Conjugating by a permutation matrix, we obtain that XI n − M
is equivalent to the matrix diag(P 1 ,... ,P n ). Hence XI n −M is equivalent

to XI n − M. From Theorem 6.3.1, M and M are similar.
Theorem 6.3.4 Let k be a field, M ∈ M n (k) a square matrix, and
P 1 ,... ,P n its similarity invariants. Then M is similar to the block-
diagonal matrix M whose jth diagonal block is the companion matrix of

P j .
The matrix M is called the first canonical form of M,or the Frobenius

canonical form of M.
Remark:If L is an extension of k (namely, a field containing k)and M ∈
M n (k), then M ∈ M n (L). Let P 1 ,... ,P n be the similarity invariants of M
as a matrix with entries in k.Then XI n −M = P diag(P 1 ,... ,P n )Q,where
P, Q ∈ GL n (k[X]). Since P, Q, their inverses, and the diagonal matrix also
belong to M n (L[X]), P 1 ,... ,P n are the similarity invariants of M as a
matrix with entries in L. In other words, the similarity invariants depend
on M but not on the field k. To compute them, it is enough to place
ourselves in the smallest possible field, namely that spanned by the entries
of M. The same remark holds true for the first canonical form. As we shall
see in the next section, it is no longer true for the second canonical form,
which is therefore less canonical.
We end this paragraph with a characterization of the minimal polyno-
mial.