Derivation of Toeplitz Matrix I'm having a difficult time understanding why the matrix for LTI systems is a Toeplitz matrix. I can see why $h_{n,m} = h_{n' + q,m' + q}$ given that $n' = n - q$ and $m' = m - q$, and $$\sum_{m'= -\infty}^\infty h_{n' + q,m' + q} x[m'] = y[n']$$ and $$\sum_{m= -\infty}^\infty h_{n,m} x[m] = y[n],$$ but the leap to the equivalence $h_{n,m} = h_{n + q,m + q}$ escapes me.
This reference should help.  If you carry out the matrix-vector multiplication, you will see they are the same. 

Also note that it is only possible to express the convolution as a matrix multiplication when the filter impulse response is of finite duration, i.e. a FIR filter. 

The equivalence $h_{n,m} = h_{n+q,m+q}$ comes from the "TI" part of LTI, that is, time-invariance. If you delay the input by $q$ samples, the output should be the same, just delayed in time by $q$.