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DETERMINANTS AND THE CAYLEY-HAMILTON THEOREM TRISTAN GIRON Throughout this note, we try to follow the same notations than in the course. F is a field, unless otherwise specified F = C. A will be a matrix in M (F) n representing the linear transformation α : V → V in a given basis of the n- dimensional F-vector space V. χ = χ shall denote its characteristic polynomial, α A and m =m its minimal polynomial. α A Afundamental result in the study of linear transformations is the Cayley-Hamilton theorem. THEOREM 1 (Cayley-Hamilton). The characteristic polynomial is annihilator. In other words, χ (α) = 0. α Exercise 2. Prove the Cayley-Hamilton theorem in the case where α is represented by a triangular matrix. B Hint: Proceed by induction. If B = (b ,...,b ) is a basis of V such that A = (α) 1 n B is triangular in B, consider V = span(b ,...,b ) for 0 ≤ k ≤ n and study the linear k 1 k transformation α −µkId, where µk = Akk, for 1 ≤ k ≤ n. A bit harder: show that this proof in fact implies the general Cayley-Hamilton as stated above. This is in essence the proof that you have been given in your course for the Cayley-Hamilton theorem. In this note we would like to give another proof of the Cayley-Hamilton theorem using more directly properties of the determinant. I. Determinants Wereview the main elements of the theory of determinants. Definition 3. Let v1,...,vk ∈ V be vectors of V, k ≥ 1. • A map φ : V ×···×V → F is k-multilinear if it is linear in every variable, i.e. for λ ∈ F,w ∈ V, and 1 ≤ j ≤ k, φ(v1,...,vj +λw,...,vk) = φ(v1,...,vj,...,vk)+λφ(v1,...,w,...,vk). • A k-multilinear map is alternated if for all permutation σ ∈ S , we have k φ(vσ(1),vσ(2),...,vσ(k)) = ε(σ)φ(v1,v2,...,vk). Here ε(σ) is the signature of the permutation σ. Date: December 13, 2019. 1 2 TRISTAN GIRON Exercise 4. It is possible that in first year you were taught that the signature of a permutation is the determinant of the permutation matrix, which would make this section circular. Let us therefore give an independent definition. 2 Let σ ∈ S , and define i(σ) to be the number of pairs (l,m) ∈ N such that 1 ≤ l < k m≤kandσ(l)>σ(m). Define ε(σ):=(−1)i(σ). • Show that for arbitrary real numbers x ,...,x ∈ R, we have 1 k Y (xσ(l)−xσ(m)) = ε(σ) Y (xl−xm). 1≤l
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