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:

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

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 $\vec{v}$:

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

Bra-Ket Notation

The dot product $\vec{u}\cdot \vec{v}$ can be written using bra-ket notation in the following way:

$$⟨u|v⟩$$

Theorem: Structure of the Real Vector Space

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

PROOF

TODO

Outer Product

Definition: Outer Product

Let $\mathbf{u}={\left[\begin{array}{ccc}{u}_1& \cdots & u_m\end{array}\right]}^{\mathsf{T}}\in {\mathbb{C}}^m$ and $\mathbf{v}={\left[\begin{array}{ccc}{v}_1& \cdots & v_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{C}}^n$ be complex column vectors.

The outer product $\mathbf{u}\otimes \mathbf{v}$ is the $m\times n$- matrix whose entry at the $i$-the row and the $j$-th column is the product of $u$‘s $i$-th component and the complex conjugate of $v$‘s $j$-th component:

$$\mathbf{u}\otimes \mathbf{v}=\left[\begin{array}{c}{u}_1\\ ⋮\\ u_m\end{array}\right]\otimes \left[\begin{array}{c}{v}_1\\ ⋮\\ v_n\end{array}\right]\stackrel{\text{def}}{=}\left[\begin{array}{cccc}{u}_1{\bar{v}}_1& u_1{\bar{v}}_2& \cdots & u_1{\bar{v}}_n\\ u_2{\bar{v}}_1& u_2{\bar{v}}_2& \cdots & u_2{\bar{v}}_n\\ ⋮& ⋮& \ddots & ⋮\\ u_m{\bar{v}}_1& u_m{\bar{v}}_2& \cdots & u_m{\bar{v}}_n\end{array}\right]$$

The outer product is equivalent to the matrix product of $\mathbf{u}$ with the Hermitian transpose of $\mathbf{v}$:

$$\mathbf{u}\otimes \mathbf{v}=\mathbf{u}{\mathbf{v}}^{†}$$

Bra-Ket Notation

The outer product $\mathbf{u}\otimes \mathbf{v}$ can be written using bra-ket notation in the following way:

$$|u⟩⟨v|$$

Theorem: Rank of the Outer Product

The outer product of two non-zero complex vectors has rank $1$.

PROOF

TODO

Theorem: Sesqulinearity of the Outer Product

The outer product is linear in the first argument:

$$(\lambda {\mathbf{u}}_1+\mu {\mathbf{u}}_2)\otimes \mathbf{v}=\lambda \cdot ({\mathbf{u}}_1\otimes \mathbf{v})+\mu \cdot ({\mathbf{u}}_2\otimes \mathbf{v})$$

The outer product is conjugate-linear in the second argument:

$$\mathbf{u}\otimes (\lambda {\mathbf{v}}_1+\mu {\mathbf{v}}_2)=\bar{\lambda }\cdot (\mathbf{u}\otimes {\mathbf{v}}_1)+\bar{\mu }\cdot (\mathbf{u}\otimes {\mathbf{v}}_2)$$

PROOF

TODO

Theorem: Hermitian Transpose of the Outer Product

The Hermitian transpose of the outer product reverses the order:

$$(\mathbf{u}\otimes \mathbf{v}{)}^{†}=\mathbf{v}\otimes \mathbf{u}$$

PROOF

TODO