# CNN - Is this a Toeplitz Matrix?

I have been reading through Chapter 9 of www.deeplearningbbook.org, where convolutional networks are being described.

The following image represents the output of a 2D convolution, without kernel flipping.

The book goes on to describe this matrix as a Toeplitz matrix where,

for univariate discrete convolution, each row of the matrix is constrained to be equal to the row above shifted by one element.

I fully understand this statement since w, x, y and z are constants in their respective columns with shifting elements.

However, there is no mention of diagonal-constants which are a key feature of such matrices. As per Wikipedia (link above) and several other sources:

a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant.

Going back to the image above, does this mean that, for instance, aw == cx == gy? How can this be ensured when all elements are different?

Authors mention the illustrative simple case of univariate discrete convolution. In univariate discrete convolution, we would be applying a 1-D kernel matrix of length $$m$$ on 1-D input data of length $$n$$. Let us take n=10 and m=3. Doing the convolution could be done by multiplying the input vector data by a n*n matrix, whose diagonal would be composed of the 3 weights of the kernel, shifted of one column to the right at each new row.