The most common [[Inner Product]] for [[Complex Numbers|Complex]] [[Vector Space|Vector Spaces]] is very similar to the real value version, except multiplication is done with the [[Complex Conjugate]]. $\huge \left< \vec A,\vec B \right> = \sum_{n} A_{i} B_{i}^{*} $ This is done to preserve the notion of [[Vector Magnitude]] / to keep magnitudes [[Real Numbers]] by defining our [[LP Norm]] as the following: $\huge \begin{align} || x | | =\left< x,x \right> \end{align} $ This is also done to preserve [[Conjugate Symmetry]] as well as positive-definiteness. $\huge \begin{align} \left< x,y \right> &= \left< y,x \right>^{*} \\ \left< x,x \right> &\geq 0 \end{align} $ ## Products with [[Roots of Unity]] $\huge \begin{align} \mat{z_{0}\\z_{1}\\\vdots } \mat{ e^{i \frac{2\pi}{N}0} \\ e^{i \frac{2\pi}{N}1} \\ \vdots } &= z_{0}+z_{1}e^{-i \frac{2\pi}{N}} + \cdots + z_{n-1}e^{-i \frac{2\pi}{N}(N-1)} \\ &= \sum_{t}^{N-1} z_{t} e^{- \frac{2\pi}{N}t} \end{align} $ Which brings us to the [[Fourier Decomposition|Discrete Fourier Transform]]. $ \huge \left< \vec z,\vec u \right> $ Where $\vec u$ is a [[Fourier Basis Vector]].