I was reading about Toeplitz matrix and found the following:
If the i,j element of A is denoted Ai,j, then we have
Ai,j = A i+1,j+1 = a i-j
So I understood that Ai,j = A i+1,j+1 which means that every member is equal to the member one row and one column after, but what about the following symbol: a i-j I don't understand it at all.
I mean each member has row index and column index but this symbol has one index.
I believe because of the structure of the Toeplitz matrix, you can assign a single index for the diagonal entries because along any given diagonal, the value is the same. For example, for a $3 \times 3$ case the pattern is:
$$ \begin{bmatrix} a_{0} & a_{-1} & a_{-2} \\ a_{1} & a_{0} & a_{-1} \\ a_{2} & a_{1} & a_{0} \\ \end{bmatrix} $$
This idea generalizes to other matrices.
More details here at wiki: https://en.wikipedia.org/wiki/Toeplitz_matrix. I hope this helps.