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SEEMOUS
2021 SEEMOUS
Problem 3
Problem 3
Part of
2021 SEEMOUS
Problems
(1)
Normal matrix
Source: 2021 SEEMOUS, P3
7/23/2021
Let
A
∈
M
n
(
C
)
A \in \mathcal{M}_n(\mathbb{C})
A
∈
M
n
(
C
)
be a matrix such that
(
A
A
∗
)
2
=
A
∗
A
(AA^*)^2=A^*A
(
A
A
∗
)
2
=
A
∗
A
, where
A
∗
=
(
A
ˉ
)
t
A^*=(\bar{A})^t
A
∗
=
(
A
ˉ
)
t
denotes the Hermitian transpose (i.e., the conjugate transpose) of
A
A
A
. (a) Prove that
A
A
∗
=
A
∗
A
AA^*=A^*A
A
A
∗
=
A
∗
A
. (b) Show that the non-zero eigenvalues of
A
A
A
have modulus one.