Matrix Notation and Dimensions

Matrices are ordered tables of numbers. To work with these mathematical structures effectively, a standard method is needed to describe their size and refer to their individual components. This involves understanding matrix dimensions and element notation.

Understanding Dimensions: Rows and Columns

The size, or dimensions, of a matrix are defined by the number of rows and the number of columns it contains. We typically say a matrix with $m$ rows and $n$ columns is an $m \times n$ matrix (read as "m by n"). The number of rows always comes first.

For example, consider this matrix $A$:

$$A = \begin{bmatrix} 5 & 12 & 6 \\ -3 & 0 & 14 \end{bmatrix}$$

Matrix $A$ has 2 rows and 3 columns, so it is a $2 \times 3$ matrix.

It's important to remember this order: rows first, then columns. A $3 \times 2$ matrix would look different, having 3 rows and 2 columns:

$$B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$$

Here, $B$ is a $3 \times 2$ matrix.

Matrices $A$ and $B$ have different dimensions. $A$ is $2 \times 3$, while $B$ is $3 \times 2$.

Referring to Matrix Elements

We often need to refer to a specific number, or element, within a matrix. We do this using indices, typically represented by subscripts. If we have a matrix $A$, the element in the $i$-th row and $j$-th column is denoted as $A_{ij}$ or $a_{ij}$. Again, the row index ($i$) comes first, followed by the column index ($j$).

Let's use matrix $A$ from before:

$$A = \begin{bmatrix} 5 & 12 & 6 \\ -3 & 0 & 14 \end{bmatrix}$$

Here's how we refer to its elements:

• The element in the 1st row and 1st column is $A_{11} = 5$.
• The element in the 1st row and 3rd column is $A_{13} = 6$.
• The element in the 2nd row and 1st column is $A_{21} = -3$.
• The element in the 2nd row and 2nd column is $A_{22} = 0$.

Notation $A_{ij}$ specifies the element at row $i$ and column $j$.

Indexing: Math vs. Programming

In mathematics and textbooks, indices usually start from 1 (1-based indexing). So, the top-left element is $A_{11}$. However, in many programming languages, including Python and its library NumPy, indexing starts from 0 (0-based indexing).

• Mathematical Notation: $A_{ij}$, where $i$ starts at 1 and $j$ starts at 1.
• NumPy/Python Notation: A[i, j], where i starts at 0 and j starts at 0.

So, the mathematical element $A_{11}$ corresponds to A[0, 0] in NumPy. Similarly, $A_{23}$ corresponds to A[1, 2]. We will primarily use the mathematical notation ($A_{ij}$) when discussing concepts, but it's essential to remember the 0-based indexing when we start implementing these ideas in Python with NumPy in the hands-on sections.

Summary of Notation

• Matrices are usually denoted by uppercase letters (e.g., $A, B, X$).
• Elements within a matrix are denoted by the corresponding lowercase letter with subscripts indicating the row and column (e.g., $a_{ij}, b_{ij}, x_{ij}$) or using the uppercase letter with subscripts ($A_{ij}$).
• The dimensions are written as rows $\times$ columns (e.g., $m \times n$).

Understanding these conventions allows us to precisely communicate about matrices and their components, which is fundamental as we move towards performing operations with them. When we work with NumPy, we'll see how its array structures directly map to these mathematical concepts, using 0-based indexing for practical access.