Understanding Endomorphisms and Eigenspaces in Linear Algebra

[borjstalker]

1. Let phi be an endomorphism such that phi^3=phi. Prove that V=E(0) (+) E(1) (+) E(-1), where E(r) denotes the eigenspace associated to the eigenvalue r.

2. Let k be a field and A a square matrix in k. Prove that if r1, r2, ..., rt are eigenvalues of A and f is an element of k[x] then f(r1), f(r2), ..., f(rt) are also eigenvalues of f(A).

 

[cristo]

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