ABSTRACT
Let T be a k-linear endomorphism of a k-vectorspace V to itself, meaning, as usual, that
T (v + w) = Tv + TW and T (cv) = c · Tv
for v, w ∈ V and c ∈ k. The collection of all such T is denoted Endk(V ), and is a vector space over k with the natural operations
(S + T )(v) = Sv + Tv (cT )(v) = c · Tv
A vector v ∈ V is an eigenvector for T with eigenvalue c ∈ k if
T (v) = c · v
or, equivalently, if (T − c · idV ) v = 0
A vector v is a generalized eigenvector of T with eigenvalue c ∈ k if, for some integer ` ≥ 1
(T − c · idV )` v = 0 We will often suppress the idV notation for the identity map on V , and just write c for the scalar operator c · idV . The collection of all λ-eigenvectors for T is the λ-eigenspace for T on V , and the collection of all generalized λ-eigenvectors for T is the generalized λ-eigenspace for T on V .
