Page 127 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
110
factorization of the jth invariant polynomial of M.Wehave thus proved
the following statement.
Theorem 6.3.6 Let Q 1 ,... ,Q s be the elementary divisors of M ∈

M n (k).Then M is similar to a block-diagonal matrix M whose diagonal
blocks are companion matrices of the Q l ’s.
The matrix M is called the second canonical form of M.

Remark: The exact computation of the second canonical form of a given
matrix is impossible in general, in contrast to the case of the first form.
Indeed, if there were an algorithmic construction, it would provide an algo-
rithm for factorizing polynomials into irreducible factors via the formation
of the companion matrix, a task known to be impossible if k = IR or CC.
Recall that one of the most important results in Galois theory, known as
Abel’s theorem, states the impossibility of solving a general polynomial
equation of degree at least five with complex coefficients, using only the
basic operations and the extraction of roots of any order.
6.3.4 Jordan Form of a Matrix
When the characteristic polynomial splits over k, which holds, for instance,
if the field k is algebraically closed, the elementary divisors have the form
r
(X − a) for a ∈ k and r ≥ 1. In that case, the second canonical form can
be greatly simplified by replacing the companion matrix of the monomial
r
(X − a) by its Jordan block
a 1 0 ··· 0
 
. . .
 . . . . 
0 . . . . . 

 .
 
J(a; r):=  . . . . . . . . . . . 0  .

 
 . . . 
 . . . . . . 1 
0 ··· ··· 0 a
r
In fact, the characteristic polynomial of J(a; r)(of size r × r)is (X − a) ,
while the matrix XI r − J(a; r) possesses an invertible minor of order r − 1,
namely
−1 0 ··· 0
 
. .
 . . . . 
 X − a . . . 
  ,
. .
 . . 
 . . 0 
X − a −1
which is obtained by deleting the first column and the last row. Again, this
shows that D n−1 (XI r − J) = 1, so that the invariant factors d 1 ,... ,d r−1
r
are equal to 1. Hence d r = D r (XI r − J)=det(XI r − J)=(X − a) .Its