16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 13

Inner Product Spaces

Mathematics — Linear Algebra · isomorphism-of-operator-spaces

Problem

Let

V

and

W

be finite-dimensional inner product spaces having the same dimension. Let

U

be an isomorphism of

V

onto

W

. Show that: (a) The mapping

T \to UTU^{-1}

is an isomorphism of the vector space

L(V,V)

onto the vector space

L(W,W)

. (b)

\text{trace}(UTU^{-1}) = \text{trace}(T)

for each

T

L(V,V)

. (c)

UT_{\alpha,\beta}U^{-1} = T_{U\alpha, U\beta}

(

T_{\alpha,\beta}

defined in Exercise 10). (d)

(UTU^{-1})^* = UT^*U^{-1}

. (e) If we equip

L(V,V)

with inner product

(T_1|T_2) = \text{trace}(T_1T_2^*)

, and similarly for

L(W,W)

, then

T \to UTU^{-1}

is an inner product space isomorphism.