Linear Algebra Done Right · Problem 5D.14

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

((a)) Give an example of a finite-dimensional complex vector space and an operator

T

on that vector space such that

T^2

is diagonalizable but

T

is not diagonalizable. ((b)) Suppose

\mathbf{F} = \mathbf{C}

k

is a positive integer, and

T \in \mathcal{L}(V)

is invertible. Prove that

T

is diagonalizable if and only if

T^k

is diagonalizable.