Euclidean space/Isometry/Structure/Fact

Then ${\displaystyle {}V}$ is an

orthogonal direct sum

${\displaystyle {}V=G_{1}\oplus \cdots \oplus G_{p}\oplus H_{1}\oplus \cdots \oplus H_{q}\oplus E_{1}\oplus \cdots \oplus E_{r}\,}$

of ${\displaystyle {}\varphi }$-invariant linear subspaces,

where the ${\displaystyle {}G_{i},H_{j}}$ are one-dimensional, and the ${\displaystyle {}E_{k}}$ are two-dimensional. The restriction of ${\displaystyle {}\varphi }$ to the ${\displaystyle {}G_{i}}$ is the identity, the restriction to ${\displaystyle {}H_{j}}$ is the negative identity, and the restriction to ${\displaystyle {}E_{k}}$ is a rotation without eigenvalue.