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2022 SEEMOUS
1
SEEMOUS 2022 Problem 1
SEEMOUS 2022 Problem 1
Source: SEEMOUS 2022
May 29, 2022
linear algebra
Matrices
sylvester inequality
eigenvalues
Problem Statement
Let
A
,
B
∈
M
n
(
C
)
A, B \in \mathcal{M}_n(\mathbb{C})
A
,
B
∈
M
n
(
C
)
be such that
A
B
2
A
=
A
B
AB^2A = AB
A
B
2
A
=
A
B
. Prove that: a)
(
A
B
)
2
=
A
B
.
(AB)^2 = AB.
(
A
B
)
2
=
A
B
.
b)
(
A
B
−
B
A
)
3
=
O
n
.
(AB - BA)^3 = O_n.
(
A
B
−
B
A
)
3
=
O
n
.
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