Isometries of Euclidean spaces

Exercise 4.1   An isometry is a linear isomorphism $\phi:(V,f)\rightarrow(W,g)$ such that $\Vert{\mathbf{u}}\Vert=\Vert\phi({\mathbf{u}})\Vert$ for all ${\mathbf{u}}\in V$. Show $\phi$ is an isometry iff $f({\mathbf{u}}_1,{\mathbf{u}}_2)=g(\phi({\mathbf{u}}_1),\phi({\mathbf{u}}_2))$ for all vectors ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ in $V$.

Exercise 4.2   Show $\phi$ is an isometry iff it maps an orthonormal basis to an orthonormal basis.

Theorem 4.3   $f({\mathbf{u}},{\mathbf{v}})=[{\mathbf{u}}]^*_{\underline{e}}[{\mathbf{v}}]_{\underline{e}}=\overline{u}_1v_1+\cdots+\overline{u}_nv_n$ where $\underline{e}$ is an orthonormal basis.

Exercise 4.4   Show that $A:V\rightarrow V$ is an isometry iff the columns of $A$ form an orthonormal basis of $\mathbb{F}^n$, where $n$ is the dimension of $V$.

Exercise 4.5   Show $A$ is an isometry iff $A^*A=I$ iff $A^*=A^{-1}$ iff $AA^*=I$. Such a matrix is called unitary.

Example 4.6   Unitary matrices living in real vector spaces are known as orthogonal matrices. The rotations and reflections of last lecture are examples under the standard inner product on $\mathbb{R}^2$.

When are diagonal matrices unitary?

If and only if $AA^*=I$ iff $\vert\lambda_i\vert=1$ for all $i$ iff all eigenvalues have unit length.

Definition 4.7   $A$ and $B$ are similar under unitary transforms, denoted $A\sim_u B$, if there exists a unitary matrix $S$ such that

$$B=S^{-1}AS=S^*AS.$$

Theorem 4.8   (Spectral Theorem)
$A=A^*$ if and only if $A$ is similar under unitary transforms to

$$% latex2html id marker 2123\left[ \begin{array}{ccc} \lamb... ...{array}\right]\mbox{ with the } \lambda_i\mbox{ real numbers.}$$