Complex Vector Space

Definition

$(V,\mathbb{C},+,\cdot )$ or $V$ is a Vector Space over complex $\mathbb{C}$

A Module whose set of scalar is a complex. $(V,+)$ is an Abelian Group scalar multiplication $\cdot$ is a Monoid action of $(\mathbb{C},\cdot )$ on $V$ the scalar multiplication is distributive with both vector and scalar addition

Complex Vector

Where we call the $\mathbf{v}\in V$ a complex Vector complex vector can be expressed as $\mathbf{v}=\mathrm{Re}(\mathbf{v})+i\mathrm{Im}(\mathbf{v})$

Complex Conjugate

Let $\mathbf{v}=(v_1,v_2,\dots ,v_n)$ $\overline{\mathbf{v}}=(\overline{v_1},\overline{v_2},\dots ,\overline{v_n})=\mathrm{Re}(\mathbf{v})-i\mathrm{Im}(\mathbf{v})$

Facts

Let $\mathbf{v},\mathbf{u}\in {\mathbb{C}}^n$ vectors $\overline{\overline{\mathbf{v}}}=\mathbf{v}$ $\overline{k\mathbf{v}}=\overline{k}\overline{\mathbf{v}}$ $\overline{\mathbf{v}\pm \mathbf{u}}=\mathbf{v}\pm \mathbf{u}$

Let $\mathbf{A}$ be $m\times k$ and $\mathbf{B}$ be $k\times n$ matrices. $\overline{\overline{\mathbf{A}}}=\mathbf{A}$ $\overline{{\mathbf{A}}^{⊺}}=(\overline{\mathbf{A}}{)}^{⊺}$ $\overline{\mathbf{AB}}=(\overline{\mathbf{A}})(\overline{\mathbf{B}})$