Linear Algebra Done Right · Problem 7A.24

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · adjoints

Problem

Suppose

T \in \mathcal{L}(V)

and

a_0 + a_1 z + a_2 z^2 + \cdots + a_{m-1} z^{m-1} + z^m

is the minimal polynomial of

T

. Prove that the minimal polynomial of

T^*

\overline{a_0} + \overline{a_1} z + \overline{a_2} z^2 + \cdots + \overline{a_{m-1}} z^{m-1} + z^m.

This exercise shows that the minimal polynomial of

T^*

equals the minimal polynomial of

T

\mathbf{F} = \mathbf{R}