Complex number

$v \in C^{n} ⟹ v = v_{1} v_{2} ⋮ v_{n}$ with $v_{i} \in C$

$\overline{(a + ib)} = a - ib$ $v^{*} = \overset{v}{ˉ}^{⊤}$

$⟨ v, w ⟩ = w^{*} v = \overline{v^{*} w}$

For $A \in C^{m \times n}$

Let $A^{*} = \overset{ˉ}{A}^{⊤}$ with $\overset{ˉ}{A} = [\overset{ˉ}{A}_{i, j} \forall_{i, j}] \in C^{m \times n}$

$(A B)^{*} = B^{*} A^{*}$

Thomas Second Brain