Eigenvalues, Eigenvectors, and Sundry Decompositions

We continue with F being an arbitrary field and V a finite dimensional vector space over F ; dimV = n. Conditions on F may be added later on. We assume T ∈ L(V ). Definition 1 We say that an element λ ∈ F is an eigenvalue of T iff there exists x ∈ V , x ̸= 0 such that Tx = λx. We call x ∈ V , x ̸= 0 such that Tx = λx an eigenvector of T , corresponding to the eigenvector λ. If λ is an eigenvector of T we call the set Eλ(T ) = {x ∈ V : Tx = λx} the eigenspace of T corresponding to λ. Notice that 0 ∈ Eλ(T ); Eλ(T ) consists of 0 and all eigenvectors of T corresponding to λ. We denote by σ(T ) the set of all eigenvalues of T .