Linear Algebra Done Right · Problem 7B.24

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Suppose

U

is a finite-dimensional vector space and

T \in \mathcal{L}(U)

. (a) Suppose

\mathbf{F} = \mathbf{R}

. Prove that

T

is diagonalizable if and only if there is a basis of

U

such that the matrix of

T

with respect to this basis equals its transpose. (b) Suppose

\mathbf{F} = \mathbf{C}

. Prove that

T

is diagonalizable if and only if there is a basis of

U

such that the matrix of

T

with respect to this basis commutes with its conjugate transpose. This exercise adds another equivalence to the list of conditions equivalent to diagonalizability in 5.55.

Related theory

The Prime Number Theorem And Dirichlet S Theorem

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