I am having trouble understanding why the system matrix of an LTI system is Toeplitz. I am following an Edx online course by Professor Richard Baraniuk of Rice University, named discrete-time signals and systems. In the course, the instructor showed that given a system matrix $H$, $H(n,m) = H(n'+q,m'+q)$ where $n'=n-q$ and $m'=m-q$, which is obvious. But, from that, he claimed the general Toeplitz structure $H(n,m)=H(n+q,m+q)$ and this is where I am getting lost.
LTI systems are related to Toeplitz matrices via convolution: The output of an LTI system is the convolution of the input with the system impulse response.
Assuming matrix equations are understood, observe how time domain convolution can be written as a matrix product, applicable to finite duration impulse responses. The resulting matrix $A$ with each column as a shifted copy of $r[n]$ is a Toeplitz matrix.