Linear Algebra Done Right · Problem 8B.2

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspaces

Problem

Suppose T∈L(V)T \in \mathcal{L}(V) is invertible. Prove that G(λ,T)=G ⁣(1λ,T−1)G(\lambda, T) = G\!\left(\frac{1}{\lambda}, T^{-1}\right) for every λ∈F\lambda \in \mathbf{F} with λ≠0\lambda \neq 0.

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