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SEEMOUS
2022 SEEMOUS
3
SEEMOUS 2022 Problem 3
SEEMOUS 2022 Problem 3
Source: SEEMOUS 2022
May 29, 2022
linear algebra
Matrices
hermitian
Inequality
Problem Statement
Let
α
∈
C
∖
{
0
}
\alpha \in \mathbb{C}\setminus \{0\}
α
∈
C
∖
{
0
}
and
A
∈
M
n
(
C
)
A \in \mathcal{M}_n(\mathbb{C})
A
∈
M
n
(
C
)
,
A
≠
O
n
A \neq O_n
A
=
O
n
, be such that
A
2
+
(
A
∗
)
2
=
α
A
⋅
A
∗
,
A^2 + (A^*)^2 = \alpha A\cdot A^*,
A
2
+
(
A
∗
)
2
=
α
A
⋅
A
∗
,
where
A
∗
=
(
A
ˉ
)
T
.
A^* = (\bar A)^T.
A
∗
=
(
A
ˉ
)
T
.
Prove that
α
∈
R
\alpha \in \mathbb{R}
α
∈
R
,
∣
α
∣
≤
2
|\alpha| \le 2
∣
α
∣
≤
2
. and
A
⋅
A
∗
=
A
∗
⋅
A
.
A\cdot A^* = A^*\cdot A.
A
⋅
A
∗
=
A
∗
⋅
A
.
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