Kunze Linear Algebra · Problem 8.3.10

Inner Product Spaces

Mathematics — Methods · adjoints-matrix-operators

Problem

Let

V

be the space of

n \times n

matrices over the complex numbers, with the inner product

(A, B) = \operatorname{tr}(AB^*)

. Let

P

be a fixed invertible matrix in

V

, and let

T_P

be the linear operator on

V

defined by

T_P(A) = P^{-1}AP

. Find the adjoint of

T_P