Page 51 - Matrices theory and applications

2. Square Matrices
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16. Let A ∈ M n(k)(k = IR or CC) be given, with minimal polynomial q.
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If x ∈ k ,the set
is an ideal of k[X], which is therefore principal.
(a) Show that I x = (0) and that its monic generator, denoted by
p x , divides q. I x := {p ∈ k[X] | p(A)x =0}
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(b) One writes r j instead of p x when x = e .Show that q is the
least common multiple of r 1 ,... ,r n .
(c) If p ∈ k[X], show that the set
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V p := {x ∈ k | p x ∈ (p)}
(the vectors x such that p divides p x )is open.
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(d) Let x ∈ k be an element for which p x is of maximal degree.
Show that p x = q. Note: In fact, the existence of an element x
such that p x equals the minimal polynomial holds true for every
field k.
17. Let k be a field and A ∈ M n×m(k), B ∈ M m×n (k)begiven.
(a) Let us define

A
XI n
M = .
B XI m
2
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Show that X m det M = X det(X I m −BA)(search for a lower
triangular matrix M such that M M is upper triangular).

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(b) Find an analogous relation between det(X I n −AB)and det M.
m
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Deduce that X P BA (X)= X P AB (X).
(c) What do you deduce about the eigenvalues of A and of B?
18. Let k be a field and θ : M n (k) → k a linear form satisfying θ(AB)=
θ(BA) for every A, B ∈ M n (k).
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(a) Show that there exists α ∈ k such that for all X, Y ∈ k ,one
T
has θ(XY )= α x j y j .
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(b) Deduce that θ = α Tr.
19. Let A n be the ring K[X 1 ,... ,X n ] of polynomials in n variables.
Consider the matrix M ∈ M n (A n ) defined by
 
1 ··· 1
 X 1 ··· X n 
 2 2 
X ··· X

 1 n  .
. .
M = 
. .
 
 . . 
X n−1 ··· X n−1
1 n
Let us denote by ∆(X 1 ,... ,X n ) the determinant of M.