Linear Algebra Done Right · Problem 5D.1

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose VV is a finite-dimensional complex vector space and T∈L(V)T \in \mathcal{L}(V). ((a)) Prove that if T4=IT^4 = I, then TT is diagonalizable. ((b)) Prove that if T4=TT^4 = T, then TT is diagonalizable. ((c)) Give an example of an operator T∈L(C2)T \in \mathcal{L}(\mathbf{C}^2) such that T4=T2T^4 = T^2 and TT is not diagonalizable.

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