What do the $i,j$ stand for when writing $A=(a_{ij})$?

Question

We often write matrices as $A=(a_{ij})$. But what do the subscripts mean? Does it simply say that $A$ is a $2$-dimensional array?

I read somewhere in the definition of the transpose of a matrix that if $A=(a_{ij})$ then its transpose is $A^T=(a_{ji})$. I had a problem with this. By $(a_{ij})$ it is understood that we are talking about the matrix $\begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\...&...&...&...\\...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}\end{bmatrix}$.

Then $(a_{ji}),(a_{mn}),(a_{pq})$ should all stand for the same matrix (the one above).

Could someone explain to me the meaning of the notation or provide a better one?

Answer 1

The first subscript, usually $i$, stands for the row index. The second one stands for the column index.

The point when writing $A^T=(a_{ji})$ is that $i$ and $j$ are not really dummy variables here as you seem to imply: what is meant is really

$$A^T=(b_{ij})\qquad \text{where}\ \ b_{ij}=a_{ji}\ \ \text{for all $i$ and $j$.}$$

$a_{ji}$ is the entry at the intersection of the $j$-th row and the $i$-th column of $A$ (you know that because here $j$ comes first), and it is meant to become the entry at the intersection of the $i$-th row and the $j$-th column in $A^T$ (you know that because in $b_{ij}$, $i$ comes first).

Answer 2

The row and the column, respectively.

In other words, $a_{ij}$ is the entry on the $i$-th row and $j$- th column of the matrix.

For instance, for the identity matrix $a_{ij}=\delta _i^j=\begin {cases} 0, i\not =j\\1, i=j\end {cases}$.

For the transpose you interchange $i$ and $j$, thus swapping rows and columns.

Answer 3

When you refer to $a_{i,j}$ it is presumed you have a table of values. $a_{5,2}$ is the value that is the second item of the fifth row. $a_{7,9}$ is the nineth value of the seventh row.

When you wrote $(a_{i,j})$ in this informal notation it means the array of placing the terms in their proper row and columns.

So $(a_{j,i})$ assumes the same values but this is the arrow were the columns are rows are transposed. That is $a_{5,2}$ was the value of the second item of the fifth row of the original matrix. It is now placed in the fifth item of the second row.

Now you might be saying (I'd be saying) Wait a minute! When were those terms ever defined in the first place? And what can I say... this is one of the reason the notation is informal.

What we should say is something like "Let $A$ be a matrix whose terms are $a_{i,j}$ for the indexes $i,j$ where the row $i$ runs from $1$ to $m$ and the column index $j$ runs from $1$ to $n$" or something like that.

Then once it's been declared we can then refer to $(a_{j,i})$ as the transposed matrix.

In that context after we have declared the matrix $(a_{i,j})$ to exists, refering to a matrix $(a_{m,n})$ or $(a_{p,q})$ wouldn't really make any clear sense. We have no idea just how the index variables $m,n$ or $p,q$ are supposed to relate.

BUT if we did the before we had defined the matrix and we declared $(a_{m,n})$ that'd just be the same as declaring $(a_{i,j})$; it's just that from here on out we'll be using the index variable labels (they are labels; nothing more) $m,n$ instead.

tl;dr

When we declare a matrix we can use the notation $(a_{i,j})$ to be any matrix with $i$ being the index variable for the row and $j$ being the index variable for the column. The is no significance to the variables or positions $i$ and $j$ and we could use any other variables in any order.

But one we have declared the matrix, all the terms $a_{i,j} $ are now fixed and any future reference to them will refer to that specific value $(a_{j,i})$ will not refer to the transpose as the row index $i$ is in the column positive and the column index $i$ will be in the row position.