Structure theorem for isometries
Let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
be an
isometry
on the
Euclidean vector space
V
{\displaystyle {}V}
Then
V
{\displaystyle {}V}
is an
orthogonal direct sum
V
=
G
1
⊕
⋯
⊕
G
p
⊕
H
1
⊕
⋯
⊕
H
q
⊕
E
1
⊕
⋯
⊕
E
r
{\displaystyle {}V=G_{1}\oplus \cdots \oplus G_{p}\oplus H_{1}\oplus \cdots \oplus H_{q}\oplus E_{1}\oplus \cdots \oplus E_{r}\,}
of
φ
{\displaystyle {}\varphi }
invariant
linear subspaces ,
where the
G
i
,
H
j
{\displaystyle {}G_{i},H_{j}}
are one-dimensional, and the
E
k
{\displaystyle {}E_{k}}
are two-dimensional. The restriction of
φ
{\displaystyle {}\varphi }
to the
G
i
{\displaystyle {}G_{i}}
is the identity, the restriction to
H
j
{\displaystyle {}H_{j}}
is the negative identity, and the restriction to
E
k
{\displaystyle {}E_{k}}
is a rotation without eigenvalue.
