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IMC
1996 IMC
3
3
Part of
1996 IMC
Problems
(1)
IMC 1996 Problem 3
Source: IMC 1996
3/5/2021
The linear operator
A
A
A
on a finite-dimensional vector space
V
V
V
is called an involution if
A
2
=
I
A^{2}=I
A
2
=
I
, where
I
I
I
is the identity operator. Let
dim
V
=
n
\dim V=n
dim
V
=
n
. i) Prove that for every involution
A
A
A
on
V
V
V
, there exists a basis of
V
V
V
consisting of eigenvectors of
A
A
A
. ii) Find the maximal number of distinct pairwise commuting involutions on
V
V
V
.
vector
linear algebra