ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5623 on D729: Conjugate symmetric complex function

Euclidean complex dot product is conjugate symmetric

Formulation 0

Let $z, w \in \mathbb{C}^{N \times 1}$ each be a D5689: Complex column matrix.

Then \begin{equation} z \cdot w = \overline{w \cdot z} \end{equation}

Formulation 1

Let $z, w \in \mathbb{C}^{N \times 1}$ each be a D5689: Complex column matrix.

Then \begin{equation} z^T \overline{w} = \overline{w^T \overline{z}} \end{equation}

Formulation 2

Let $z, w \in \mathbb{C}^{N \times 1}$ each be a D5689: Complex column matrix.

Then \begin{equation} \sum_{n = 1}^N z_n \overline{w}_n = \overline{\sum_{n = 1}^N w_n \overline{z}_n} \end{equation}

Proofs

Proof 0

Let $z, w \in \mathbb{C}^{N \times 1}$ each be a D5689: Complex column matrix.

Using results

(i)	R5298: Complex conjugatation is an additive operation
(ii)	R3429: Complex conjugation is a multiplicative operation
(iii)	R2950: Complex conjugation operation is an involution

we have \begin{equation} \sum_{n = 1}^N z_n \overline{w}_n = \sum_{n = 1}^N \overline{\overline{z}_n w_n} = \overline{\sum_{n = 1}^N \overline{z}_n w_n} \end{equation} $\square$