Complex Vectors

Definition: Complex Column Vector

A complex column vector is a column vector $\vec{v}\in {\mathbb{C}}^n$ over the complex numbers.

Definition: Complex Row Vector

A complex row vector is a row vector $\vec{v}\in {\mathbb{C}}^{1\times n}$ over the complex numbers.

Dot Product

Definition: Complex Dot Product

The dot product of two complex column vectors $\vec{u}={\left[\begin{array}{ccc}{u}_1& \cdots & u_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{C}}^n$ and $\vec{v}={\left[\begin{array}{ccc}{v}_1& \cdots & v_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{C}}^n$ is defined as:

$$\vec{u}\cdot \vec{v}\stackrel{\text{def}}{=}\sum _{k=1}^n\overline{u_k}{v}_k$$

TIP

The dot product $\vec{u}\cdot \vec{v}$ of two complex column vectors is equivalent to the matrix product of the Hermitian transpose of $\vec{u}$ with $\vec{v}$:

$$\vec{u}\cdot \vec{v}={\vec{u}}^{†}\vec{v}$$

Theorem: Structure of the Real Vector Space

The dot product is an inner product on $({\mathbb{C}}^n,\mathbb{C},+,\cdot )$.

PROOF

TODO