Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 719

21.5.1.A Example of a maximizing unitary matrix

Consider a function of the form

P− 1

1
1+[ U † AU ] kk

φ ( U )= (21 . 50)

k =0

where A is a fixed P P Hermitian positive semidefinite matrix, and U is a

unitary matrix to be chosen such that φ ( U ) is maximized. Here the quantities

×

d k =[ U † AU ] kk

denote the diagonal elements of U † AU . First observe that

d k =Tr[ U † AU ]=Tr[ UU † A ]=Tr[ A ] ,

(21 . 51)

k

which is fixed, independent of U . Moreover d k ≥ 0 because A is positive definite.

If U is chosen to diagonalize A then [ U † AU ] kk = λ k (eigenvalues of A ). From
Ex. 21.9 we know that

1
1+ λ k ≥

1
1+ d k

k k

Summarizing, the unitary matrix U that maximizes φ ( U ) is the one that diag-
onalizes A . The maximized objective function is given by
φ ( U )=
k

1
1+ λ k

21.5.1.B Example of a minimizing unitary matrix

Now imagine that our goal is to minimize rather than maximize φ ( U )inEq.
(21.50) by choice of the unitary matrix U . Then what is the best U ?Sincethe
average value c = k d k /P is independent of U (by Eq. (21.51)) it follows from
Lemma 21.1 that

d P− 1 ] T 1] T [ d 0 d 1 ... c [1 1 ...

no matter how U is chosen. Since

P− 1

1
1+ x i

f ( x )=

i =0

is Schur-convex in x for x i ≥ 0 (from Eq. (21.39)), it follows that

P− 1 P− 1

1
1+[ U † AU ] kk ≥

1
1+ c =

P
1+ c

k =0 k =0