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SEEMOUS
2019 SEEMOUS
2
Traces of matrices
Traces of matrices
Source: SEEMOUS 2019, problem 2
March 18, 2019
linear algebra
college contests
Problem Statement
Let
A
1
,
A
2
,
…
,
A
m
∈
M
n
(
R
)
A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})
A
1
,
A
2
,
…
,
A
m
∈
M
n
(
R
)
. Prove that there exist
ε
1
,
ε
2
,
…
,
ε
m
∈
{
−
1
,
1
}
\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}
ε
1
,
ε
2
,
…
,
ε
m
∈
{
−
1
,
1
}
such that:
t
r
(
(
ε
1
A
1
+
ε
2
A
2
+
⋯
+
ε
m
A
m
)
2
)
≥
t
r
(
A
1
2
)
+
t
r
(
A
2
2
)
+
⋯
+
t
r
(
A
m
2
)
\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2)
tr
(
(
ε
1
A
1
+
ε
2
A
2
+
⋯
+
ε
m
A
m
)
2
)
≥
tr
(
A
1
2
)
+
tr
(
A
2
2
)
+
⋯
+
tr
(
A
m
2
)
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