# How to read $(a_{ij})^{n}_{i,j=1}$ in matrix definition

In an assignment given in a numerical analysis class, the following notation is used to define an $n\times n$ matrix:

$$A = (a_{ij})^{n}_{i,j=1} = \left\lbrace \begin{array}{ll}0, & i>j,\\1, & i=j,\\j-i+1,&i<j\\\end{array}\right.$$

The very left and very right part is perfectly clear to me. But I do not understand what the $(\ldots)_{i,j=1}$ part is supposed to mean. I only know $a_{ij}$ to denote the element at the position $ij$.

So what does the subscript $i,j=1$ and superscript $n$ mean here?

It is just a notation to denote that $A$ is an $n\times n$ matrix with coefficients $a_{ij}$, for $i$ and $j$ ranging from $1$ to $n$.
That is, the notation $A=(a_{ij})_{1\leq i,j\leq n}$, or $A=(a_{ij})_{i,j=1}^n$, identifies the matrix $A$ to the tuple of its $n^2$ coefficients $(a_{11}, a_{12}, \dots, a_{1n}, a_{21}, \dots, a_{nn})$.
Then, the RHS is actually an abuse of notation: it is the definition of $a_{ij}$ (for any fixed $1\leq i,j\leq n$), not of $A=(a_{ij})_{ij}$. A correct definition should read
$A = (a_{ij})^{n}_{i,j=1}$, where for $1\leq i,j\leq n$ we have $$a_{ij} = \left\lbrace \begin{array}{ll}0, & i>j,\\1, & i=j,\\j-i+1,&i<j\\\end{array}\right.$$