Linear Algebra Done Right · Problem 5D.5

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose

V

is a finite-dimensional complex vector space and

T \in \mathcal{L}(V)

. Prove that

T

is diagonalizable if and only if

V = \operatorname{null}(T - \lambda I) \oplus \operatorname{range}(T - \lambda I)

for every

\lambda \in \mathbf{C}