Since we only have a finite number of \( \alpha \) and \( \beta \) values which are non-zero, we can rewrite the above convolution expressions as a matrix-vector multiplication
$$ \boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ \alpha_1 & \alpha_0 & 0 & 0 \\ \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ 0 & 0 & \alpha_2 & \alpha_1 \\ 0 & 0 & 0 & \alpha_2 \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. $$The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). In this case we have
$$ \boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ \beta_1 & \beta_0 & 0 \\ \beta_2 & \beta_1 & \beta_0 \\ \beta_3 & \beta_2 & \beta_1 \\ 0 & \beta_3 & \beta_2 \\ 0 & 0 & \beta_3 \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. $$Note that the use of these matrices is for mathematical purposes only and not implementation purposes. When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. We rather code the convolutions in the minimal memory footprint that they require.
Does the number of floating point operations change here when we use the commutative property?
The above matrices are examples of so-called Toeplitz matrices. A Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant. For instance the last matrix, which we rewrite as
$$ \boldsymbol{A}=\begin{bmatrix}a_0 & 0 & 0 \\ a_1 & a_0 & 0 \\ a_2 & a_1 & a_0 \\ a_3 & a_2 & a_1 \\ 0 & a_3 & a_2 \\ 0 & 0 & a_3 \end{bmatrix}, $$with elements \( a_{ii}=a_{i+1,j+1}=a_{i-j} \) is an example of a Toeplitz matrix. Such a matrix does not need to be a square matrix. Toeplitz matrices are also closely connected with Fourier series discussed below, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. The example above shows that we can represent linear convolution as multiplication of a Toeplitz matrix by a vector.