Conjugate symmetry inner product

1. What is conjugate symmetry inner product?
2. Why is inner product conjugate symmetric?
3. Why there is a conjugate in the inner product?
4. What is conjugate symmetry?

What is conjugate symmetry inner product?

By the conjugate symmetry we also have (w, 0) = 0. Lemma 2. The inner product is anti-linear in the second slot, that is, (u, v + w) = (u, v) + (u, w) for all u, v, w ∈ V and (u, av) = a(u, v).

Why is inner product conjugate symmetric?

An inner-product space may be defined over both real complex planes. Remember for a real vector space V, the conjugates of vectors a,b in V are just a,b themselves. So if you are using V to define your inner-product space conjugate symmetry is just symmetry ⟨a,b⟩=⟨b,a⟩.

Why there is a conjugate in the inner product?

The conjugate is necessary because you want to define a norm ‖⋅‖:V→R≥0 by using that inner product, putting ‖x‖=√⟨x,x⟩, and for this you need ⟨x,x⟩ to be real. The conjugation gives ⟨x,x⟩=¯⟨x,x⟩∈R. Save this answer.

What is conjugate symmetry?

Conjugate symmetry is an entirely new approach to symmetric Boolean functions that can be used to extend existing methods for handling symmetric functions to a much wider class of functions. These are functions that currently appear to have no symmetries of any kind. Conjugate symmetries occur widely in practice.