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

Result R5624 on D723: Symmetric map

Euclidean real dot product is symmetric

Formulation 0

Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then \begin{equation} x \cdot y = y \cdot x \end{equation}

Formulation 1

Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then \begin{equation} x^T y = y^T x \end{equation}

Formulation 2

Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then \begin{equation} \sum_{n = 1}^N x_n y_n = \sum_{n = 1}^N y_n x_n \end{equation}

Proofs

Proof 0

Let $x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.