Multiplying a Vector by a Matrix

An important operation in the context of matrices is the product between a matrix and a vector. Take some matrix

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \\ \end{bmatrix}.

This matrix can be subdivided into its columns. For this, the idea of seeing a matrix as a vector of vectors is very useful. I introduced this idea in the introductory lesson about matrices.

Let us divide $A$ into columns:

A = \begin{bmatrix} \vec{a}_1 & \vec{a}_2 & \vec{a}_3 \end{bmatrix}.

Here, $\vec{a}_1$ is the first column of $A$ , $\vec{a}_2$ is the second, and $\vec{a}_3$ is the third and final column. Therefore, we can say that

\vec{a}_1 = \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \\ a_{41} \end{bmatrix}, \vec{a}_2 = \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \\ a_{42} \end{bmatrix}, \vec{a}_3 = \begin{bmatrix} a_{13} \\ a_{23} \\ a_{33} \\ a_{43} \end{bmatrix}.

Keep this in mind.

We now have another vector $\vec{x}$ whose number of components is the same as the number of columns in our matrix (in our example, $3$ ):

\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.

We can now perform a multiplication between the matrix $A$ and the vector $\vec{x}$ . The product is the linear combination of the columns of $A$ , where the coefficients come from the corresponding element of $\vec{x}$ . In other words

\begin{align*} A \vec{x} &= x_1 \vec{a}_1 + x_2 \vec{a}_2 + x_3 \vec{a}_3 \\ &= x_1 \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \\ a_{41} \end{bmatrix} + x_2 \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \\ a_{42} \end{bmatrix} + x_3 \begin{bmatrix} a_{13} \\ a_{23} \\ a_{33} \\ a_{43} \end{bmatrix} \\ &= \begin{bmatrix} x_1 a_{11} \\ x_1 a_{21} \\ x_1 a_{31} \\ x_1 a_{41} \end{bmatrix} + \begin{bmatrix} x_2 a_{12} \\ x_2 a_{22} \\ x_2 a_{32} \\ x_2 a_{42} \end{bmatrix} + \begin{bmatrix} x_3 a_{13} \\ x_3 a_{23} \\ x_3 a_{33} \\ x_3 a_{43} \end{bmatrix} \\ &= \begin{bmatrix} x_1 a_{11} + x_2 a_{12} + x_3 a_{13} \\ x_1 a_{21} + x_2 a_{22} + x_3 a_{23} \\ x_1 a_{31} + x_2 a_{32} + x_3 a_{33} \\ x_1 a_{41} + x_2 a_{42} + x_3 a_{43} \\ \end{bmatrix} = \vec{b}. \end{align*}

As you can see, the product of a matrix and a vector is another vector. I chose the letters $A$ , $\vec{x}$ , and $\vec{b}$ very deliberately, since these letters are conventionally used in this context (this operation will be very important for future lessons).

Dimensions of the Operands and the Result

As you can infer from the example above, the matrix used was a $4 \times 3$ matrix. The vector we used in the multiplication had $3$ components.

This is very important: the product of a matrix and a vector is only defined for matrix-vector pairs where the number of components in the vector is the same as the number of columns in the matrix. This, for instance, means that you cannot multiply a vector with $5$ components with a matrix of size $5 \times 3$ , because that matrix would only have $3$ columns.

More formally, if $A$ is an $m \times n$ matrix, for the operation $A\vec{x}$ to be defined, $\vec{x}$ must have $n$ components.

The resulting vector $\vec{b}$ has the same number of components as the number of rows in the matrix involved in the operation. For an $m \times n$ matrix, the resulting vector $\vec{b}$ will have $m$ components.

Example

Allow me to demonstrate through the following numerical example. Take the matrix

A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \\ 4 & -2 & 1 \end{bmatrix},

and the vector

\vec{x} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.

We compute $A\vec{x}$ :

\begin{align*} A\vec{x} &= \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \\ 4 & -2 & 1 \end{bmatrix} \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} \\ &= 2 \begin{bmatrix} 1 \\ -1 \\ 2 \\ 4 \end{bmatrix} + (-1) \begin{bmatrix} 0 \\ 3 \\ 1 \\ -2 \end{bmatrix} + 3 \begin{bmatrix} 2 \\ 1 \\ 0 \\ 1 \end{bmatrix} \\ &= \begin{bmatrix} 2 \\ -2 \\ 4 \\ 8 \end{bmatrix} + \begin{bmatrix} 0 \\ -3 \\ -1 \\ 2 \end{bmatrix} + \begin{bmatrix} 6 \\ 3 \\ 0 \\ 3 \end{bmatrix} \\ &= \begin{bmatrix} 8 \\ -2 \\ 3 \\ 13 \end{bmatrix}. \end{align*}

Thus,

\vec{b} = \begin{bmatrix} 8 \\ -2 \\ 3 \\ 13 \end{bmatrix}.