FFT
When one is dealing with complex numbers , there are a few adjustments that one need to make for the following central ideas

Length : It is not transpose of a vector times vector. It is conjugate transpose of a vector times a vector
$latex \displaystyle \text{ Inner product} = \overline y^T \; y \; = y^H \; y &fg=000000$
where $latex {H}&fg=000000$ stands for Hermitian

Symmetric matrix condition:
$latex \displaystyle \overline A^T = A^H = A &fg=000000$

Perpendicularity :
$latex \displaystyle q_i^H = \overline q_i^T \times q_j = 0 , \forall i \neq j &fg=000000$

orthogonal becomes unitary
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FFT is based on a method that makes multiplication of a matrix quicker. It cuts down the order of computation from $latex {n^2}&fg=000000$ to $latex {{1\over2} n \log n}&fg=000000$