Lemma 2. For a positive semi-definite Hermitian matrix A ∈ C M × M \mathbf{A}\in\mathbb{C}^{M\times M} A∈CM×M, the condition Rank ( A ) = 1 \left(\mathbf{A}\right)=1 (A)=1 is equivalent to t h e following conditions the\textit{ following conditions} the following conditions
(34a)
max B T r ( A B ) − 2 v − T r ( V ) ≥ 0 , T r ( B ) = 1 V − A + v I M ⪰ 0 M , B ⪰ 0 M , V ⪰ 0 M , \max_{\mathbf{B}}Tr\left(\mathbf{AB}\right)-2v-Tr\left(\mathbf{V}\right)\geq0,Tr\left(\mathbf{B}\right)=1\\\mathbf{V}-\mathbf{A}+v\mathbf{I}_M\succeq\mathbf{0}_M,\mathbf{B}\succeq\mathbf{0}_M,\mathbf{V}\succeq\mathbf{0}_M, BmaxTr(AB)−2v−Tr(V)≥0,Tr(B)=1V−A+vIM⪰0M,B⪰0M,V⪰0M,
( 34 b ) (34b) (34b)
where v v v and the Hermitian matrices B , V ∈ C M × M are \mathbf{B} , \mathbf{V} \in \mathbb{C} ^{M\times M}\textit{ are} B,V∈CM×M are
t h e introduced auxiliary variables. the\textit{ introduced auxiliary variables. } the introduced auxiliary variables.
