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.
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