MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 410

since a k > 0 . The lower bound on the right-hand side can therefore be achieved
if we choose the optimal U f to be the unitary matrix that diagonalizes H † H
completely and arranges diagonal elements as in Eq. (12.44). The proof of the
above lemma is therefore complete once we prove Eq. (12.45):

Proof of Eq. (12.45). The proof is based on a beautiful theorem on Hermi-
tian matrices which we first review (see Horn and Johnson [1985], p. 189).
Recall first that all eigenvalues of a Hermitian matrix are real. Let A be a
P × P Hermitian matrix and let A M be any M × M principal submatrix,

that is, a matrix obtained by deleting any P M rows and the corresponding

columns from A . Let the eigenvalues of A be arranged as

−

λ 0 ( A ) ≥ λ 1 ( A ) ...≥ λ P− 1 ( A ) (12 . 47)

and let the eigenvalues of A M be arranged as

λ 0 ( A M ) ≥ λ 1 ( A M ) ...≥ λ M− 1 ( A M ) . (12 . 48)

Then the eigenvalues λ k ( A M ) are interleaved between the eigenvalues λ k ( A )
as follows:

λ 0 ( A ) ≥ λ 0 ( A M ) ≥ λ L ( A ) ,

λ 1 ( A ) ≥ λ 1 ( A M ) ≥ λ L +1 ( A ) ,

.
λ M− 1 ( A )

≥ λ M− 1 ( A M ) ≥ λ P− 1 ( A ) ,

where L = P M . This is demonstrated in Fig. 12.4. Now consider any

unitary matrix U f which has performed the partial diagonalization (12.37).
The eigenvalues of the M

−

M leading principal submatrix of U f H † HU f are
clearly μ k , whereas the eigenvalues of U f H † HU f are just σ h,k (eigenvalues
of H † H ) regardless of U f . The interlacing property

×

σ h,k ≥ σ h,k + L , μ k ≥ 0 ≤ k ≤ M − 1 , (12 . 49)

therefore holds. This implies in particular that Eq. (12.45) is true.

12.4.3 Finding the optimal Σ f

With the optimal unitary U f shown to be the matrix that diagonalizes H † H ,
we now turn to the computation of the optimal coe cients a k in Eq. (12.38)
which determine the diagonal elements of Σ f . For this, recall again the neces-
sary condition (12.32) for optimality derived earlier based on stationarity of the
Lagrangian. This condition is reproduced below:

( F † H † HF ) − 1 = η F † F ,

(12 . 50)