Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 720

Moreover, there is a particular choice of U that achives d k = c for all k ,thatis,

⎡ ⎤

c × ... ×

⎣ × c. . . × ⎦

U † AU = . (21 . 52)

. . .

. . .

×× ... c

So the unitary U which minimizes φ ( U ) is the one that makes the diagonal
elements of U † AU identical.

Proof of existence. To prove the existence of a unitary U which achieves Eq.
(21.52), observe that if V is any un ita ry matrix with identical magnitudes
for all its elements (i.e.,

=1 / √ P for all ( k, m )), then, for any diagonal

| V km |

matrix Λ ,

P− 1 P− 1

P
m

1

[ V † ΛV ] kk = [ V † ] km λ m [ V ] mk = V mk | 2 =

λ m | λ m = c,

m =0 m =0

and this is independent of k . Thus, given arbitrary Hermitian A ,ifwe
choose U = TV , where T diagonalizes A , and V is as above, then

[ U † AU ] kk =[ V † T † ATV ] kk =[ V † ΛV ] kk = c.

Such a U therefore minimizes φ ( U ) . There are many examples of unitary
matrices V which satisfy

1
√ P | V km | = for all ( k, m ) .

One example is the normalized DFT matrix W , which has

[ W ] km = e −j 2 πkm/P √ P

Another example is the Hadamar d matrix, which is a real orthogonal matrix
with elements 1 (divided by √ P for normalization). Hadamard matrices

exist for certain values of P, e.g., when P is a power of two [Moon and
Stirling, 2000].

±

Here is the summary of what we have shown:

♠ Theorem 21.7. A minimization problem with unitary U . Consider the

function

P− 1

1
1+[ U † AU ] kk

φ ( U )= (21 . 53)

k =0

where A is a fixed P P Hermitian positive semidefinite matrix. Then the

minimum value of φ ( U )as U varies over the set of all unitary matrices is given
by

×

P
1+ c

φ ( U )=