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SEEMOUS
2008 SEEMOUS
Problem 3
Problem 3
Part of
2008 SEEMOUS
Problems
(1)
FE Mn(R)->[n], inequality f(XY)≤min{f(X),f(Y)}
Source: SEEMOUS 2008 P3
6/17/2021
Let
M
n
(
R
)
\mathcal M_n(\mathbb R)
M
n
(
R
)
denote the set of all real
n
×
n
n\times n
n
×
n
matrices. Find all surjective functions
f
:
M
n
(
R
)
→
{
0
,
1
,
…
,
n
}
f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}
f
:
M
n
(
R
)
→
{
0
,
1
,
…
,
n
}
which satisfy
f
(
X
Y
)
≤
min
{
f
(
X
)
,
f
(
Y
)
}
f(XY)\le\min\{f(X),f(Y)\}
f
(
X
Y
)
≤
min
{
f
(
X
)
,
f
(
Y
)}
for all
X
,
Y
∈
M
n
(
R
)
X,Y\in\mathcal M_n(\mathbb R)
X
,
Y
∈
M
n
(
R
)
.
inequalities
functional equation
fe
matrix
linear algebra