16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.5.9

Operators on Inner Product Spaces

Mathematics — Linear Algebra · conjugation-operators

Problem

Let

V

be the inner product space in Exercise 7 and

G

the group of unitary matrices in

V

. If

B

is in

G

, let

C_B

denote the linear operator on

V

defined by

C_B(A) = BAB^{-1}.

Show that (a)

C_B

is a unitary operator on

V

; (b)

C_{B_1B_2} = C_{B_1}C_{B_2}

; (c) there is no unitary transformation

U

V

such that

UL_BU^{-1} = C_B

for all

B

G