Euclidean spaces

Let $\mathbb{F}$ denote $\mathbb{C}$ or $\mathbb{R}$.

Definition 3.1   A bilinear map $f:V\times V\rightarrow\mathbb{F}$ is a Hermitian form if

• for fixed ${\mathbf{u}}$, $F({\mathbf{u}},\bullet):V\rightarrow\mathbb{F}$ is linear
• $f({\mathbf{u}},{\mathbf{v}})=\overline{f({\mathbf{v}},{\mathbf{u}})}$.

Example 3.2   Define $A^*=\overline{A}^t$. A matrix $A$ is Hermitian if $A^*$=$A$. Given such a matrix, then $f({\mathbf{u}},{\mathbf{v}})={\mathbf{u}}^*A{\mathbf{v}}$ is a Hermitian form.

Exercise 3.3   Show that any Hermitian form can be written as above for some Hermitian matrix $A$.

Definition 3.4   Let $Q_f({\mathbf{u}})=f({\mathbf{u}},{\mathbf{u}}):V\rightarrow\mathbb{R}$. (Why is this always real?) This is a quadratic form.

• $Q_f$ is positive semidefinite if for all ${\mathbf{u}}$, $Q_f({\mathbf{u}})\geq0$.
• $Q_f$ is positive definite if for all ${\mathbf{u}}\neq0$, $Q_f({\mathbf{u}})>0$.

Definition 3.5   A Euclidean space is a pair $(V,f)$ consisting of a vector space $V$ (over $\mathbb{C}$ or $\mathbb{R}$) and a positive definite Hermitian form $f$.

Example 3.6   $\mathbb{R}^2$ with the usual dot product ${\mathbf{a}}\cdot{\mathbf{b}}=\vert{\mathbf{a}}\vert\cdot\vert{\mathbf{b}}\vert\cos\theta$, where $\theta$ is the angle between ${\mathbf{a}}$ and ${\mathbf{b}}$ is an example; this is the same as ${\mathbf{a}}\cdot{\mathbf{b}}=a_1b_1+a_2b_2$.

Definition 3.7   ${\mathbf{u}}$ is perpendicular to ${\mathbf{v}}$, denoted by ${\mathbf{u}}\perp{\mathbf{v}}$, if $f({\mathbf{u}},{\mathbf{v}})=0$. We define the norm of ${\mathbf{u}}$ by $\Vert{\mathbf{u}}\Vert=\sqrt{f({\mathbf{u}},{\mathbf{u}})}$.

Definition 3.8   An orthonormal basis is a basis ${\mathbf{e}}_1,\ldots,{\mathbf{e}}_n$ such that $f({\mathbf{e}}_i,{\mathbf{e_j}})=\delta_{ij}$.

Theorem 3.9   Every finite-dimensional Euclidean space has an orthonormal basis.

Expressing a vector in coordinates involves solving an inhomogeneous system of equations. However if we wish to express ${\mathbf{u}}$ in orthonormal coordinates

$${\mathbf{u}}=\alpha_1{\mathbf{e}}_1+\cdots+\alpha_n{\mathbf{e}}_n$$, we calculate $$\alpha_i=f({\mathbf{e}}_i,{\mathbf{u}}).$$