Inner Product

An inner product is an operator that maps a pair of vectors from a vector space to a real number. In the real vector space, ${\mathbb{R}}^n$, the inner product is the dot product. An inner product can only be defined on a vector space fixed over the field $\mathbb{R}$ or $\mathbb{C}$.

The inner product of $\vec{u}$ and $\vec{v}$ is notated as:

$$⟨\vec{u},\vec{v}⟩$$

A vector space defined with an inner product is called an inner product space. Note that this can only happen if the vector space, $V$ is fixed over the field $\mathbb{R}$ or $\mathbb{C}$.

Definition §

Let $\vec{u},\vec{v},\vec{w}$ be different vectors in $V_{\mathbb{F}}$ fixed over the field $\mathbb{F}=\mathbb{R}\text{ or }\mathbb{C}$. Let $j,k\in \mathbb{R}$ be a real scalar.For an inner product, $⟨\vec{u},\vec{v}⟩$ to be defined, it must satisfy the following properties:

1. Symmetry of Inner Product §

The operands in the inner product can be swapped and produce the same result.

$$⟨\vec{u},\vec{v}⟩=⟨\vec{v},\vec{u}⟩$$

2. Linearity of Scalar Multiplication §

A scalar can be ‘factored out’ of either operand.

$$⟨k\vec{u},j\vec{v}⟩=kj⟨\vec{u},\vec{v}⟩$$

3. Linearity of Vector Addition §

A vector addition can be ‘separated’ out into another inner product.

$$⟨\vec{u}+\vec{v},\vec{w}⟩=⟨\vec{u},\vec{w}⟩+⟨\vec{v},\vec{w}⟩$$

4. Positivity §

The inner product of the same vector, with itself, must always be greater than zero, unless it operates on a zero vector.

$$\{\begin{array}{ll}⟨\vec{u},\vec{u}⟩=0,& \vec{u}=\vec{0}\\ ⟨\vec{u},\vec{u}⟩>0,& \text{else}\end{array}$$

Hermitian Inner Product §

A Hermitian inner product operates over the complex vector space ${\mathbb{C}}^n$ and returns a complex number.

It satisfies the same properties as a normal inner product except symmetry is defined differently:

Hermitian Symmetry §

$$⟨\vec{u},\vec{v}⟩=\overline{⟨\vec{v},\vec{u}⟩}$$

The bar represents the complex conjugate.

Matrix Representation §

An inner product operating over a real vector space, ${\mathbb{R}}^n$ can be expressed as the product of coordinate vectors and a symmetric matrix:

Matrix Representation of an Inner Product

$$⟨\vec{u},\vec{v}⟩=[\vec{u}{]}^T\cdot A\cdot [\vec{v}]$$

$[\vec{u}],[\vec{v}]$ refer to coordinate vectors with respect to the standard basis in R.