Page 123 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
106
6.3.1 Example: The Companion Matrix of a Polynomial
Given a polynomial
P(X)= X + a 1 X
there exists a matrix B ∈ M n (k) such that the list of invariant factors of
the matrix XI n − B is (1,... , 1,P). We may take the companion matrix
associated to P to be n n−1 + ··· + a n ,
0 ··· ··· 0 −a n
 
. . .
 . . . 
1 . . .
 
 
. . . .

 . . . .  .
0 . . . .
B P := 
 . . . . 

 . . . . . . 0 . . 

0 ··· 0 1 −a 1
Naturally, any matrix similar to B P would do as well, because if B =
Q −1 B P Q,then XI n −B is similar, hence equivalent, to XI n −B P .In order
to show that the invariant factors of B P are the polynomials (1,... , 1,P),
we observe that XI n −B P possesses a minor of order n−1 that is invertible,
namely, the determinant of the submatrix
 
−1 X 0 ··· 0
. . .
 . . . . . 
0 . . . . 

.
 
. . . .  .
 . . . 
 . . . . 0
 
.
. . .
 . . 
 . . . X 
0 ··· ··· 0 −1
We thus have D n−1 (XI n − B P ) = 1, so that the invariant factors
d 1 ,... ,d n−1 are all equal to 1. Hence d n = D n (XI n − B P )=det(XI n −
B P ), the characteristic polynomial of B P ,namely P.
In this example P is also the minimal polynomial of B P .Infact,if Q is
a polynomial of degree less than or equal to n − 1,
Q(X)= b 0 X n−1 + ··· + b n−1 ,
1
the vector Q(A)e reads
1
n
b 0 e + ··· + b n−1e .
Hence Q(A) = 0 and deg Q ≤ n − 1imply Q = 0. The minimal polynomial
is thus of degree at least n. It is thus equal to the characteristic polynomial.
6.3.2 First Canonical Form of a Square Matrix
Let M ∈ M n (k)beasquare matrix and P 1 ,... ,P n ∈ k[X] its similarity
invariants. The sum of their degrees n j (1 ≤ j ≤ n)is n. Let us denote