Indexing of a Matrix

Just like the elements of a vector $\vec{v}$ are referred to as $v_1, v_2, \dots, v_n$, where $n$ is the number of dimensions of the vector, there is a way to refer to elements of a matrix as well.

Referring to Elements of a Matrix

If we have a matrix $A$ (capital $A$), we refer to its elements as $a_{ij}$ (lowercase $a$), where $i$ refers to the row the element finds itself in, and $j$ is its column.

Therefore, the first row of a matrix will have elements $a_{11}, a_{12}, \dots, a_{1n}$, where $n$ is the number of columns in the matrix.

The first column will have elements $a_{11}, a_{21}, \dots, a_{m1}$, where $m$ is the number of rows in the matrix.

For instance, a general $3 \times 3$ matrix $A$ would look like this:

You can also use an uppercase $A$ (like $A_{ij}$). You could say they are interchangeable, but it can also be a bit more subtle: $A_{ij}$ explicitly refers to the entry of matrix $A$ at row $i$ and column $j$, whereas $a_{ij}$ denotes the more generic element in the matrix. They really refer to the same entry, but the uppercase version emphasizes the matrix while the lowercase version emphasizes its elements.

For you to better understand the subtle difference, here is an example. We have a matrix

As you can see, I used lowercase $a$s to denote the elements in the matrix. But if I refer to a standalone element, I would use $A_{ij}$.

Sometimes, for convenience when typing on a keyboard, the notation $A(i,j)$ is used to refer to an element of a matrix $A$. The same idea applies: $i$ is the row, $j$ is the column.

For example, take the matrix

Then

• $a_{11} = 2$
• $a_{12} = 4$
• $a_{21} = 1$
• $a_{22} = 7$

Specifying the Size of a Matrix

When we refer to some matrix $A$ through its capital letter name, we can specify the size as an index: $A_{m \times n}$.

For instance, if the matrix $A$ has size $3 \times 4$, we can refer to it as $A_{3 \times 4}$, if we deem it to be important information in the current context.

Referring to an Entire Row or Column

We can use the notation $A_{i \star}$ to refer to the $i$th row. Similarly, we can use $A_{\star j}$ to refer to the $j$th column.

Sometimes, the notation $A_{i \cdot}$ and $A_{\cdot j}$ is used, but I will stick to the first version.

For instance, take the matrix

Then

and

$A_{2 \star}$ is said to be a row vector, and $A_{\star 3}$ is said to be a column vector.

I have not mentioned this in the chapter about vectors, but we have only ever worked with column vectors this far. Now you know that there is actually a difference between writing a vector as a column or row.

This concept is well-understood with the idea that I have touched upon in the chapter page about matrices. You can see vectors as a matrix with only one column. Basically, a column vector is a matrix of size $m \times 1$, and a row vector is a matrix of size $1 \times n$.