Page 54 - Matrices theory and applications

24. Let P ∈ k[X] be a polynomial of degree n that splits completely in
k.Let B P be the companion matrix


.
.
.
.


.
1



.

.
.
.
.
.
 .


0
.
B P := 

.

 . 0 ··· . . ··· . 0 . . . . . . −a n 2.9. Exercises 37
.
.
 . . . . . . 0 . . 

0 ··· 0 1 −a 1
Find a matrix H ∈ M n (k), whose transpose is of Vandermonde type,
such that
HB P = diag(λ 1 ,... ,λ n )H.
This furnishes a direct proof of the fact that when the roots of P are
simple, B P is diagonalizable.
25. (E. Formanek [14])
Let k be a field of characteristic 0.
(a) Show that for every A, B, C ∈ M 2 (k),
2
[A, B] ,C =0.
Hint: use the Cayley–Hamilton theorem.
(b) Show that for every M, N ∈ M 2 (k),
MN + NM − Tr(M)N − Tr(N)M+
(Tr(M)Tr(N) − Tr(MN))I 2 =0.
One may beginwiththe case M = N and recognize a classical
theorem, then “bilinearize” the formula.
(c) If π ∈ S r (S r is the symmetric group over {1,... ,r}), one
r
defines a map T π : M 2 (k) → k in the following way. One de-
composes π as a product of disjoint cycles, including the cycles
of order one, which are the fixed points of π:
) ··· .
)(b 1 ,... ,b k 2
π =(a 1 ,... ,a k 1
One sets then
) ···
)Tr(N b 1
T π (N 1 ,... ,N r )= Tr(N a 1
··· N b k 2
··· N a k 1
(note that the right-hand side depends neither on the order of
the cycles in the product nor on the choice of the first index
inside each cycle, because of the formula Tr(AB)=Tr(BA)).
Show that for every N 1 ,N 2 ,N 3 ∈ M 2 (k), one has

(π)T π (N 1 ,N 2 ,N 3 )= 0.
π∈S 3