Inner Product

An inner product is a measure of similarity.

Description

Given a linear space, it may be possible to define a binary operation that describes how similar two elements are. This is called an inner product and is generally notated as ⟨a, b⟩ for any a and b in that space.

An inner product must satisfy the following properties:

• ⟨a, b⟩ = ⟨b, a⟩

• ⟨a + z, b⟩ = ⟨a, b⟩ + ⟨z, b⟩

• ⟨y * a, b⟩ = y * ⟨a, b⟩

  • Note that this specifically is linearity in the first argument. In some contexts, linearity in the second argument is preferred. Either is sufficient for defining an inner product. This mostly only comes up when there are complex values.

• ⟨a, a⟩ ≥ 0 unless a is the zero vector

  • recall that linearity requires the existence of a zero vector

In Euclidean space (Rⁿ), the dot product is an inner product.

Other notable inner products and the corresponding spaces are:

• Frobenius product for matrices (R_{m x n})

  • Essentially, flatten a matrix into a column and calculate the dot product.

• Definite integrals for the given range on the real line (C[a,b])

• Evaluation inner products for polynomials (P_n)

  • For P₂ space, let polynomials p and q be p(x) = a₁x² + a₂x + a₃ and q(x) = b₁x² + b₂x + b₃.

  • The evaluation inner product is a₁*b₁ + a₂*b₂ + a₃*b₃.

If an inner product can be defined for a linear space, then it is an inner product space.

Hermitian Inner Product

A Hermitian inner product is defined for the complex vector space.

A Hermitian inner product must satisfy the following properties:

• 

• ⟨a + z, b⟩ = ⟨a, b⟩ + ⟨z, b⟩

• ⟨y * a, b⟩ = y̅ * ⟨a, b⟩

• ⟨a, a⟩ ≥ 0 unless a is the zero vector

In Cⁿ space, the Hermitian inner product is given by a₁b̅₁ + ... + a_nb̅_n.

CategoryRicottone