Is there a mathematical notation of indexing a matrix? Do matrices in linear algebra support an operation of indexing them analogous to array indexing?
For example:
$$
A =
\left [\begin{array}{cc}
    1 & 2 \\
    3 & 4
\end{array}\right]
$$
In C, a fixed 2 by 2 array in 32 bit integer space could be described as:
int32_t A[][2] = {
    {1, 2}, 
    {3, 4}
};

and allows operations like:
A[0][0] = 2;

Is there a universally accepted equivalent to such an operation in linear algebra?
I understand that a Matrix is not a C Array equivalent, but I am curious whether such an operation is supported.
 A: Try $ e_i^t Ae_j = a_{ij} $ where $ e_i $ the i-th Standard Vector.
$e_i = (0,…,0,1,0,….0)^t$ and the $1$ ist the i-th Element.
A: $A_{ij}$ is a common notation for the $(i,j)$th entry of a matrix $A$.
$i$ specifies the row, $j$ specifies the column, and the indexes start with $1$, rather than $0$.
So if $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then
$A_{11} = 1$, $A_{12} = 2$, $A_{21} = 3$, and $A_{22} = 4$.
A: The most common one is $A_{ij} $, although you will often $a_{ij} $ in lowercase. Slightly less common is $A_{i,j} $ (usually when the indices are themselves a formula). Far less common is $A (i,j) $; sometimes used when there are other subindices/superindices around.
A: No, quite independently of matrices there is no standard mathematical notation for mutating operations, because there is no notion of executing a mathematical text. If we wrote a hypothetical formula with a side effect, what would make that side effect actually happen? We are free to reread our formulas if we don't quite understand them the first time around; would that make the side effect happen a twice or more?
However, it is possible to describe mathematical values in en operational way, as "the result of taking such and so value and performing this and that operation on it". Note that the meaning of such a description does not change by "executing" it, since it specifies which value to start with. Things like bringing matrices into echelon form are often described using this kind of description, but it usually does not use a symbolic formalism to describe just what operations are to be performed.
One case that is somewhat of this nature is evaluating polynomials by writing say $P(2)$ (although I personally prefer the more explicit $P[X:=2]$), which means (more or less) the result of taking the expression referred to by $P$ and replacing every instance of $X$ by the value $2$, and then performing the arithmetic operations (addition, multiplications, powers,...).
But for modifying a matrix value in a specific way, I don't think there is any standardised notation.
