# 2. RELATIONSHIP BETWEEN BOTH FORMULATIONS OF THE ц UPPER BOUND

The reader is referred to chapter 5 (subsection 2.3) for a presentation of both formulations of the mixed ц upper bound of (Fan et al., 1991; Young et al., 1995). The following Lemma is extracted from (Young et al., 1995). It is worth pointing out that the transformations in the Lemma below (from (D, G) to (D, G) and from (D, G) to (D, G)) do not depend on the complex matrix M.

LEMMA 2..1 Let:

with F = IN + G2. Assume that D € D and G Є Q satisfy equation

(10.1) . Let:

D = P

G = W*D-iGD~kr

where the unitary matrix U and the hermitian positive definite matrix P result from the polar decomposition:

Then D Є D and G Є G satisfy equation (10.2). Conversely, assume that D Є D and G Є G satisfy equation (10.2). Let:

D = DF~2D G = pDGF-tD

Then D Q D and D Q Q satisfy equation (10.1).

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