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⟩

• Linearity: ⟨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.

• ⟨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⟩

• Linearity: ⟨y * a, b⟩ = y * ⟨a, b⟩

• Conjugate linearity: ⟨a, y * 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