Euclidean space/Isometry/Structure/Fact

Structure theorem for isometries

Let

$\phi :V⟶V$

be an isometry on the Euclidean vector space $V$.

Then $V$ is an

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 $\phi$-invariant linear subspaces,

where the $G_i,H_j$ are one-dimensional, and the $E_k$ are two-dimensional. The restriction of $\phi$ 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