16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.4.3

Inner Product Spaces

Mathematics — Linear Algebra · unitary-operators-complex-numbers

Problem

Let

V

be the set of complex numbers, regarded as a real vector space. (a) Show that

(\alpha|\beta) = \operatorname{Re}(\alpha\bar{\beta})

defines an inner product on

V

. (b) Exhibit an (inner product space) isomorphism of

V

onto

R^2

with the standard inner product. (c) For each

\gamma

V

, let

M_\gamma

be the linear operator on

V

defined by

M_\gamma(\alpha) = \gamma\alpha

. Show that

(M_\gamma)^* = M_{\bar{\gamma}}

. (d) For which complex numbers

\gamma

M_\gamma

self-adjoint? (e) For which

\gamma

M_\gamma

unitary? (f) For which

\gamma

M_\gamma

positive? (g) What is

\det(M_\gamma)

? (h) Find the matrix of

M_\gamma

in the basis

\{1, i\}

. (i) If

T

is a linear operator on

V

, find necessary and sufficient conditions for

T

to be an

M_\gamma

. (j) Find a unitary operator on

V

which is not an

M_\gamma