Conjugate Symmetry

The property of any field with a conjugate to remain symmetrical from inner products. $⟨\beta |\alpha ⟩=\overline{⟨\alpha |\beta ⟩}$

Example

1. $\beta =(2+i,3-i)$, $\alpha =(1-i,2+i)$
2. $⟨\beta |\alpha ⟩={\sum }_{i=1}^n{\beta }_i{\overline{\alpha }}_i=(2+i)(1+i)+(3-i)(2-i)=8-2i$
3. $\overline{⟨\alpha |\beta ⟩}=\overline{{\sum }_{i=1}^n{\alpha }_i{\overline{\beta }}_i}=\overline{(1-i)(2-i)+(2+i)(3+i)}=\overline{8+2i}=8-2i$