Complex Inner Product Spaces

Inner product: defines the norm

provides a way to measure angles and length
introduces concept of orthogonality

Dot Product

Real dot product

u = (u_{1}, u_{2}, \dots, u_{n}) u \cdot v = u_{1} v_{1} + u_{2} v_{2} + \dots + u_{n} v_{n}

v = (v_{1}, v_{2}, \dots, v_{n})

E.g. $u = (1, 1, - 1, - 1)$ , $v = (1, - 1, 1, - 1)$

Compute $u \cdot v$ .

u \cdot v = 1 \times 1 + 1 \times (- 1) + (- 1) \times 1 + (- 1) \times (- 1) = 1 - 1 - 1 + 1 = 0

For vectors $u$ and $v \in R^{n}$

∥ v ∥^{2} = v \cdot v

∥ u - v ∥^{2} = (u - v) \cdot (u - v)

v \cdot v = v_{1} \cdot v_{1} + v_{2} \cdot v_{2} + \dots + v_{n} v_{n} = v_{1}^{2} + v_{2}^{2} + \dots + v_{n}^{2} = ∥ v ∥^{2}

By replacing $u - v$ :

(u - v) \cdot (u - v) = ∥ u - v ∥^{2}

Given vectors $u$ , $v$ and $w$ in $R^{n}$ and scalar $k$ in $R$ , the following algebraic identities apply

u \cdot v = v \cdot u (commutative)

u \cdot (v + w) = u \cdot v + u \cdot w (Left Distributive)

(u + v) \cdot w = u \cdot w + v \cdot w (Right Distributive)

k u \cdot v = k (u \cdot v) (Left Homogeneity)

u \cdot (k v) = k (u \cdot v) (Right Homogeneity)

Given $u = (u_{1}, u_{2}, \dots, u_{n})$ and $v = (v_{1}, v_{2}, \dots, v_{n})$

u \cdot v = u_{1} v_{1} + u_{2} v_{2} + \dots + u_{n} v_{n} = v_{1} u_{1} + v_{2} u_{2} + \dots + v_{n} u_{n} = v \cdot u

Complex Dot Products

In $C^{n}$ , the definition of the complex dot product differs its real counterpart

Given two vectors $u = (u_{1}, u_{2}, \dots, u_{n})$ , $v = (v_{1}, v_{2}, \dots, v_{n})$ in $C$ , the complex dot product of $u$ and $v$ is defined as:

u \cdot v = u_{1}^{*} v_{1} + u_{2}^{*} v_{2} + \dots + u_{n}^{*} v_{n} = i = 1 \sum n u_{i}^{*} v_{i}

E.g $u = (1, i)$ and $v = (i, - i)$

= u \cdot v = 1 \times 1 + - i \times i = - 1 + i

Dot product in $C^{n}$ is usually a complex scalar