# How can the convolution operation be implemented as a matrix-vector multiplication?

How can the convolution operation used by CNNs be implemented as a matrix-vector multiplication? We often think of the convolution operation in CNNs as a kernel that slides across the input. However, rather than sliding this kernel (e.g. using loops), we can perform the convolution operation "in one step" using a matrix-vector multiplication, where the matrix is a circulant matrix containing shifted versions of the kernel (as rows or columns) and the vector is the input.

How exactly can this operation be performed? I am looking for a detailed step-by-step answer that shows how the convolution operation (as usually presented) can be performed using a matrix-vector multiplication.

Is this the usual way the convolution operations are implemented in CNNs?