16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.3.3

Operators on Inner Product Spaces

Mathematics — Linear Algebra · adjoint-general-inner-product

Problem

Let

V

be the space of

n \times 1

matrices over

C

, with the inner product

(X|Y) = Y^*GX

(where

G

is an

n \times n

matrix such that this is an inner product). Let

A

be an

n \times n

matrix and

T

the linear operator

T(X) = AX

. Find

T^*

. If

Y

is a fixed element of

V

, find the element

Z

V

which determines the linear functional

X \to Y^*X

. In other words, find

Z

such that

Y^*X = (X|Z)

for all

X

V