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

Result R5296 on D999: Complex matrix
Subresult of R5295: Complex conjugate of a product with complex matrix and a complex column matrix

Complex conjugate of a product of two complex column matrices

Formulation 0

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

Then \begin{equation} \overline{z^T w} = \overline{z}^T \overline{w} \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)	R5297: Complex conjugate of a product of two complex numbers

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