Linear Algebra Done Right · Problem 5D.8

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose VV is finite-dimensional and T∈L(V)T \in \mathcal{L}(V). Let λ1,…,λm\lambda_1, \ldots, \lambda_m denote the distinct nonzero eigenvalues of TT. Prove that dim⁡E(λ1,T)+⋯+dim⁡E(λm,T)≤dim⁡range⁡T.\dim E(\lambda_1, T) + \cdots + \dim E(\lambda_m, T) \leq \dim \operatorname{range} T.

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