I know "Matrix" is usually defined as any array of numbers or other mathematical objects arranged in rows and columns. But the problem is that seems not really like a rigorous mathematical definition. Cause some notions used in the definition like "Array" or "Arrangement" are ambiguous. I mean Matrix is an important concept which we use it to define other things like operators, vectors, tensors, etc, but Matrix itself is not well defined.
Can we define Matrix in terms of more fundamental notions like sets, n-tuples, maps, functions, etc? Or should we take it as a primitive notion?
Thank you in advance.
You certainly could. For instance, for $m \in \mathbb N$ write $[m]$ for the set $\{1,\ldots,m\}$. Then, you could formally define an $n \times m$ matrix with entries in (say) $\mathbb R$ as a function $[n] \times [m] \to \mathbb R$. Then if $A : [n] \times [m] \to \mathbb R$ is a matrix, you do not write $A(i,j)$ for the $(i,j)$-th entry, but instead $A_{i,j}$.
After using this formal definition, you mostly stop thinking of $A$ as a function to avoid confusion, since $A$ also represents a linear mapping $\mathbb R^n \to \mathbb R^m$, which is really the more important thing here. But formally, it still is a function.