16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9

Inner Product Spaces

Mathematics — Linear Algebra · unitary-operators-inner-products

Problem

Let

V

be a finite-dimensional inner product space and

T

a positive linear operator on

V

. Let

p_T

be the inner product on

V

defined by

p_T(\alpha, \beta) = (T\alpha|\beta)

. Let

U

be a linear operator on

V

and

U^*

its adjoint with respect to

(\,|\,)

. Prove that

U

is unitary with respect to the inner product

p_T

if and only if

T = U^*TU