Euclidean space/Isometry/Structure/Fact

Structure theorem for isometries

Let

$\varphi \colon V\longrightarrow V$

be an isometry on the euclidean vector space ${}V$.

Then ${}V$ is a

orthogonal direct sum

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

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

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

Proof, Write another proof