4.3a Matrix-Vector Multiplication

Matrix-Vector Multiplication

A matrix $A\in {\mathbb{R}}^{m\times n}$ and a vector ${\mathbb{R}}^n$ can be multiplied to create a new vector $\vec{b}\in {\mathbb{R}}^m$,

A \vec{x}= \begin{bmatrix} A_{11} & A_{12} & \dots & A_{1n} \\ A_{21} & A_{22} & ... & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m1} & A_{m2} & ... & A_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n

\end{bmatrix}

=
\begin{bmatrix}
A_{11}x_1 + A_{12}x_2 + \ldots + A_{1n}x_n \
A_{21}x_1 + A_{22}x_2 + \ldots + A_{2n}x_n \
\vdots \
A_{m1}x_1 + A_{m2}x_2 + \ldots + A_{mn}x_n \
\end{bmatrix} = \vec{b},

where each element of the resulting vector $b_i = \sum^{n}_{j=1}A_{ij}x_j$ equals the [[3.4b Outer Product|outer product]] of the row vectors of the matrix and the factor vector.

Explanation

A smart way of interpreting matrix-vector multiplication is expressing it as a linear combination of the column vectors of $A$ and the respective $i$th element of $\vec{x}$:

$$A\vec{x}=x_1\left[\begin{array}{c}{A}_{11}\\ A_{21}\\ ⋮\\ A_{m1}\end{array}\right]+x_2\left[\begin{array}{c}{A}_{12}\\ A_{22}\\ ⋮\\ A_{m2}\end{array}\right]+\dots +x_n\left[\begin{array}{c}{A}_{1n}\\ A_{2n}\\ ⋮\\ A_{mn}\end{array}\right].$$

Example

Multiplying a random matrix by a Standard Unit Vector

Let’s think about the following equation,

$$\left[\begin{array}{cc}3& 6\\ 2& 0.5\\ 1& -3\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=0\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]+1\left[\begin{array}{c}6\\ 0.5\\ -3\end{array}\right]=\left[\begin{array}{c}6\\ 0.5\\ -3\end{array}\right].$$

Multiplying a matrix by a standard unit vector essentially “selects” a column from the matrix.