Matrix–Vector Product

36. Matrix–Vector Product

A special case of matrix multiplication: a matrix times a single column vector.

For an $𝑚 \times 𝑛$ matrix $𝐴$ and a vector $\vec{𝑥} \in ℝ^{𝑛}$ , the product $𝐴 \vec{𝑥} \in ℝ^{𝑚}$ has entries:

{(𝐴 \vec{𝑥})}_{𝑖} = \sum_{𝑗 = 1}^{𝑛} 𝑎_{𝑖 𝑗} 𝑥_{𝑗}

Example

𝐴 = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] \vec{𝑥} = [\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]

𝐴 \vec{𝑥} = [\begin{matrix} 1 \cdot 7 + 2 \cdot 8 + 3 \cdot 9 \\ 4 \cdot 7 + 5 \cdot 8 + 6 \cdot 9 \end{matrix}] = [\begin{matrix} 50 \\ 122 \end{matrix}]

1. Row view (dot products): each entry is a dot product of a row of $𝐴$ with $\vec{𝑥}$ .

2. Column view (linear combination): $𝐴 \vec{𝑥}$ is a linear combination of $𝐴$ ’s columns, weighted by the entries of $\vec{𝑥}$ :

𝐴 \vec{𝑥} = 𝑥_{1} {\vec{𝑎}}_{1} + 𝑥_{2} {\vec{𝑎}}_{2} + \dots + 𝑥_{𝑛} {\vec{𝑎}}_{𝑛}

where ${\vec{𝑎}}_{𝑗}$ is the $𝑗$ -th column of $𝐴$ .

This view explains why $𝐴 \vec{𝑥} = \vec{𝑏}$ has a solution iff $\vec{𝑏}$ lies in the column space of $𝐴$ .

Every matrix defines a linear transformation $𝑇_{𝐴} : ℝ^{𝑛} \to ℝ^{𝑚}$ :

𝑇_{𝐴} (\vec{𝑥}) = 𝐴 \vec{𝑥}

Conversely, every linear transformation between finite-dimensional spaces is a matrix–vector product (after fixing bases).

𝐴 (\vec{𝑥} + \vec{𝑦}) = 𝐴 \vec{𝑥} + 𝐴 \vec{𝑦}

𝐴 (𝑐 \vec{𝑥}) = 𝑐 (𝐴 \vec{𝑥})

$𝐴$ applied to the zero vector returns the zero vector.