Isometry/C/Diagonalizability/Section

Lemma

Let ${}\varphi \colon V\rightarrow V$ be a linear isometry on a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$, endowed with an inner product. Let ${}U\subseteq V$ denote an invariant linear subspace. Then also the orthogonal complement ${}U^{\perp }$ is invariant. In particular, ${}\varphi$ can be written as a direct sum

${}\varphi =\varphi _{U}\oplus \varphi _{U^{\perp }}\,,$

where the restrictions ${}\varphi _{U}$ and ${}\varphi _{U^{\perp }}$

are also isometries.

Proof

We have

${}U^{\perp }={\left\{v\in V\mid \left\langle v,u\right\rangle =0{\text{ for all }}u\in U\right\}}\,.$

For such a ${}v\in U^{\perp }$ and an arbitrary ${}u\in U$, we have

${}\left\langle \varphi (v),u\right\rangle =\left\langle \varphi ^{-1}(\varphi (v)),\varphi ^{-1}(u)\right\rangle =\left\langle v,u'\right\rangle =0\,,$

since ${}u'=\varphi ^{-1}(u)\in U$ due to the invariance of ${}U$. Therefore, ${}\varphi (v)\in U^{\perp }$.

$\Box$

The following statement is called spectral theorem or, more specific, spectral theorem for complex isometries.

Theorem

Let ${}V$ be a finite-dimensional ${}\mathbb {C}$-vector space, endowed with an inner product, and let

$\varphi \colon V\longrightarrow V$

be an isometry. Then ${}V$ has an orthonormal basis of eigenvectors of ${}\varphi$. In particular, ${}\varphi$ is

diagonalizable.

Proof

We do induction on the dimension of ${}V$. The statement is clear in the one-dimensional case. Due to the Fundamental theorem of algebra and fact, ${}\varphi$ has an eigenvalue and an eigenvector, which we can normalize. Let ${}E\subseteq V$ be the corresponding eigenline. Since we have an isometry, the orthogonal complement ${}E^{\perp }$ is also, due to fact, ${}\varphi$-invariant, and the restriction

$\varphi {|}_{E^{\perp }}\colon E^{\perp }\longrightarrow E^{\perp }$

is again an isometry. By the induction hypothesis, there exists on ${}E^{\perp }$ an orthonormal basis consisting of eigenvectors. Together with the first eigenvector, these form an orthonormal basis of eigenvectors of ${}V$.

$\Box$