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Eigenvalues and Eigenvectors 12 One of the central problems in linear algebra is this: given a vector space V finitely generated over a field F , and given an endomorphism α of V , is there a way to select a basis B of V so that the matrix ΦBB (α) is as nice as possible? In this chapter, we will begin by defining some basic notions which will help us address this problem. Let V be a vector space over a field F and let α ∈ End(V ). A scalar c ∈ F is an eigenvalue of α if and only if there exists a vector v = 0V satisfying α(v) = cv . Such a vector is called an eigenvector1 of α associated with the eigenvalue c. Thus we see that a nonzero vector v ∈ V is an eigenvector of α if and only if the subspace F v of V is invariant under α . Every eigenvector of α is associated with a unique eigenvalue of α but any eigenvalue has, as a rule, many eigenvectors associated with it. The set of all eigenvalues of α is called the spectrum of α and is denoted by spec(α). Thus, c ∈ spec(α) if and only if the endomorphism cσ1 − α of V is not monic. Example If V is a vector space over R and if α ∈ End(V ) satisfies α2 = −σ , then 1 spec(α) = ∅. To see this, note that if v is an eigenvector corresponding to an eigen- value c then −v = α2 2 2 2 (v) = c v and so (c + 1)v = 0V , implying that c = −1, which is impossible for a real number c. In particular, if α ∈ End(R2) is defined by α : a → −b then spec(α) = ∅. b a 1The terms “eigenvalue” and “eigenvector” are due to Hilbert. Eigenvalues and eigenvectors are sometimes called characteristic values and characteristic vectors, respectively, based on termi- nology used by Cauchy. Sylvester coi