Conjugate transpose inner product

1. Why there is a conjugate in the inner product?
2. What is a matrix multiplied by its conjugate transpose?
3. How do you conjugate transpose?

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 a matrix multiplied by its conjugate transpose?

If a matrix is equal to its own conjugate transpose, it is said to be self-adjoint and is called a Hermitian.

How do you conjugate transpose?

Definition: If is an matrix with entries from the field , then the Conjugate Transpose of is obtained by taking the complex conjugate of each entry in and then transposing .