Matrices

35. Matrices

$𝑚 \times 𝑛$ matrix $𝑨$

$𝑚$ : rows
$𝑛$ : columns

𝑨 = [\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑚 1} & 𝑎_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}]

35.1. Matrix-Vector Products

𝑨 = [\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑚 1} & 𝑎_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}]

𝐱 = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}]

𝑨 𝐱 = [\begin{matrix} 𝑎_{11} 𝑥_{1} + 𝑎_{12} 𝑥_{2} + \dots + 𝑎_{1 𝑛} 𝑥_{𝑛} \\ 𝑎_{21} 𝑥_{1} + 𝑎_{22} 𝑥_{2} + \dots + 𝑎_{2 𝑛} 𝑥_{𝑛} \\ ⋮ \\ 𝑎_{𝑚 1} 𝑥_{1} + 𝑎_{𝑚 2} 𝑥_{2} + \dots + 𝑎_{𝑚 𝑛} 𝑥_{𝑛} \end{matrix}] = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}]

For the dot product to be defined, the number of columns in the matrix $𝑨$ (which is $𝑛$ ) must match the number of elements in the vector $\vec{𝑥}$ (also $𝑛$ ).

The result of multiplying matrix $𝑨$ and vector $\vec{𝑥}$ will be a column vector with dimensions $𝑚 \times 1$ , where $𝑚$ is the number of rows in the matrix $𝑨$

(𝑚 \times 𝑛) \cdot (𝑛 \times 1) = 𝑚 \times 1

As Row vectors

\begin{matrix} 𝐚 = [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ ⋮ \\ 𝑎_{𝑛} \end{matrix}] \\ 𝐛 = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}] \end{matrix}

\begin{matrix} 𝐚^{𝑇} = [𝑎_{1}, 𝑎_{2}, \dots, 𝑎_{𝑛}] \\ 𝐛^{𝑇} = [𝑏_{1}, 𝑏_{2}, \dots, 𝑏_{𝑛}] \end{matrix}

𝑨 = [\begin{matrix} [𝑎_{1}, 𝑎_{2}, \dots, 𝑎_{𝑛}] \\ [𝑏_{1}, 𝑏_{2}, \dots, 𝑏_{𝑛}] \end{matrix}]

𝑨 = [\begin{matrix} 𝐚 \\ 𝐛 \end{matrix}]

𝐱 = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}]

[\begin{matrix} 𝐚^{𝑇} \\ 𝐛^{𝑇} \end{matrix}] \cdot 𝐱 = [\begin{matrix} 𝐚 \cdot 𝐱 \\ 𝐛 \cdot 𝐱 \end{matrix}]

As Column Vectors

\begin{matrix} \vec{𝑎} = [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ ⋮ \\ 𝑎_{𝑛} \end{matrix}] \\ \vec{𝑏} = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}] \end{matrix}

𝑨 = [\begin{matrix} [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ ⋮ \\ 𝑎_{𝑛} \end{matrix}] [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}] \end{matrix}]

𝑨 = [\begin{matrix} \vec{𝑎} & \vec{𝑏} \end{matrix}]

\vec{𝑥} = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \end{matrix}]

𝑨 \vec{𝑥} = 𝑥_{1} \vec{𝑎} + 𝑥_{2} \vec{𝑏}