Isometry/Decomposition/Section

Lemma

Let ${}V\neq 0$ be a real finite-dimensional vector space, and let

$\varphi \colon V\longrightarrow V$

denote an endomorphism. Then there exists a ${}\varphi$-invariant linear subspace ${}U\subseteq V$ of dimension

${}1$ or ${}2$.

Proof

We may assume ${}V=\mathbb {R} ^{n}$, and that ${}\varphi$ is described by the matrix ${}M$ with respect to the standard basis. If ${}\varphi$ has an eigenvalue, then we are done. Otherwise, we consider the corresponding complex mapping, that is,

$\varphi \colon \mathbb {C} ^{n}\longrightarrow \mathbb {C} ^{n},$

which is given by the same matrix ${}M$. This matrix has a complex eigenvalue ${}a+b{\mathrm {i} }$, and a complex eigenvector ${}v\in \mathbb {C} ^{n}$. In particular, we have

${}Mv=(a+b{\mathrm {i} })v\,.$

Writing

${}v=v_{1}+{\mathrm {i} }v_{2}\,$

with ${}v_{1},v_{2}\in \mathbb {R} ^{n}$, this means

${}Mv_{1}+{\mathrm {i} }Mv_{2}=Mv=(a+b{\mathrm {i} }){\left(v_{1}+{\mathrm {i} }v_{2}\right)}=av_{1}-bv_{2}+{\mathrm {i} }(av_{2}+bv_{1})\,.$

Comparing the real part and the imaginary part we can deduce that ${}Mv_{1},Mv_{2}\in \langle v_{1},v_{2}\rangle$. Therefore, the real linear subspace ${}\langle v_{1},v_{2}\rangle$ is invariant.

$\Box$

Theorem

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

We do induction over the dimension ${}n$ of ${}V$. The one-dimensional case is clear, due to fact. Let ${}n=2$. The determinant has, because of fact either the value ${}1$ or ${}-1$. In case it is ${}-1$, the characteristic polynomial has two distinct zeroes, and these zeroes are, due to fact, ${}1$ and ${}-1$. Then we have a reflection at an axis, and

${}V=G\oplus H\,.$

If the determinant is ${}1$, then we are in the situation of fact, and we have a rotation. If the angle of rotation is ${}0$, then we have the identity, and we can decompose ${}V=G_{1}\oplus G_{2}$. If the angle of rotation is ${}\pi$, then we have a point reflection ${}-\operatorname {Id}$; therefore, we have a decomposition ${}V=H_{1}\oplus H_{2}$. For all other angles, there is no eigenvector.

Let now ${}n\geq 1$ be arbitrary, and suppose that the statement is proven for smaller dimensions. Because of fact, there exists a ${}\varphi$-invariant linear subspace ${}U$ of dimension ${}1$ or ${}2$, and, because of fact, there exists an invariant orthogonal complement, that is,

${}V=U\oplus W\,.$

The induction hypothesis, applied to ${}W$, yields the result.

$\Box$

In this decomposition, ${}G_{1}\oplus \cdots \oplus G_{p}$ is the eigenspace for the eigenvalue ${}1$, and ${}H_{1}\oplus \cdots \oplus H_{q}$ is the eigenspace for the eigenvalue ${}-1$; the decompositions are not unique. The Isometry is proper if and only if ${}q$ is even.