Hermitian Dot product

Definition 2.3.1. Sia V uno spazio vettoriale su ℂ. Una forma sesquilineare su V è un'applicazione g: V x V ⟶ ℂ tale che

g(v1 + v2) = g(v1, w),  g(v, w1 + w2) = g(v, w1 + g(v, w2),
g(cv, w) = cg(v,w),   g(v, cw) = c̄g(v,w)

per tutti i v,w, v1, v2, w1, w2 ∈ V e c ∈ ℂ.  □

An inner product on the complex vector space V is a map that takes two vectors v,w ∈ V and produces a complex number

<−,−>: ℂn × ℂn ⟶ ℂ

⟨ ( x 1 , … , x n ) , ( y 1 , … , y n ) ⟩ = ∑ i = 1 n x i ¯ y i

subject to the following requirements, for u,v,w ∈ V and c,d ∈ ℂ:

  1. Sesquilinerarity (linear in its 1st argument and only semilinear in its 2nd argument):

    ⟨cu + dv, w⟩ = c ⟨u,w⟩ + d ⟨v,w⟩
    ⟨u, cv + dw⟩ = c̄ ⟨u,v⟩ + d̄ ⟨u,w⟩

    for the 2nd argument, the c and the d on the righthand side get conjugated and are not preserved (if c and d were real then the Hermitian product would be bilinear).

  2. Conjugate Symmetry (Hermitian symmetry):

    ⟨v,w⟩ = ⟨v,w⟩

  3. Positivity:

    ||v||2 = ⟨v,w⟩ ≥ 0 and ⟨v,v⟩ = 0 if and only if v = 0

The canonical Euclidean dot product is bilinear meaning it is linear in both arguments, symmetric and positive definite. The Hermitian dot product is sesquilinear (meaning half-linear), complex linear in the second argument and complex antilinear in the first argument, conjugate symmetric (exchanging the 1st and 2nd argument changes the scalar product by complex conjugation) and positive definite.

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