A matrix is a rectangular array of numbers.

Each number is called an entry or element of the matrix.

🔢 Examples

General matrix:

$$ \begin{bmatrix} 4 & 3 & 0 & 6 & -1 & 0 \\ 0 & 2 & -4 & -7 & 1 & 3 \\ -6 & 1 & 5 & -12 & 0 & 1 \end{bmatrix} $$

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📏 Matrices

The size of a matrix is denoted by $n \times m$, where:

• $n$ = number of rows
• $m$ = number of columns

Type	Shape	Example
General Matrix	$n \times m$	$\begin{bmatrix} 4 & 3 & 0 & 6 & -1 & 0 \\ 0 & 2 & -4 & -7 & 1 & 3 \\ -6 & 1 & 5 & -12 & 0 & 1 \end{bmatrix}$
Square Matrix	$n \times n$	$\begin{bmatrix} 7 & 10 \\ 8 & 0 \end{bmatrix}$
Column Matrix	$n \times 1$	$\begin{bmatrix} 4 \\ 2 \\ -1 \\ 9 \\ 5 \end{bmatrix}$
Row Matrix	$1 \times m$	$\begin{bmatrix} 1 & -5 \end{bmatrix}$
Scalar Matrix	$1 \times 1$	$[-2]$

Often when dealing with 1 x 1 matrices we will drop the surrounding brackets and just write -2.

⚠️

What vector means Note that sometimes column matrices and row matrices are called column vectors and row vectors respectively

We do need to be careful with the word vector however as in later chapters the word vector will be used to denote something much more general than a column or row matrix

🟦 Square Matrices & Main Diagonal

The main diagonal of a square matrix consists of entries:

$$ a_{11},\ a_{22},\ \dots,\ a_{nn} $$

These run from the top-left to bottom-right of the matrix:

$$ \begin{bmatrix} \color{blue}{a_{11}} & a_{12} & \cdots & a_{1n} \\ a_{21} & \color{blue}{a_{22}} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & \color{blue}{a_{nn}} \end{bmatrix} $$

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🧾 Entry Notation

To refer to a specific entry in a matrix $A$, use: $a_{ij}$ or $(A)_{ij}$

• $i$ = row index
• $j$ = column index

General Use Case Upper case letters are generally used to refer to matrices (e.g., $A$, $B$, $C$)

Lower case letters generally are used to refer to numbers/entries (e.g., $a_{ij}$, $b_{ij}$)

🧰 General Matrix Representation

A general $n \times m$ matrix $A$ can be written as:

🔣 Subscript Notation

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1m} \\ a_{21} & a_{22} & \cdots & a_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} \end{bmatrix} $$

…with size denoted

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1m} \\ a_{21} & a_{22} & \cdots & a_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} \end{bmatrix}_{n \times m} $$

Note We don’t generally subscript the size of the matrix as we did in the second case, but on occasion it may be useful to make the size clear.

🧭 Vector Notation

• Column matrix (vector): $\mathbf{a}$ or $\vec{a}$ $$ \mathbf{\vec{a}} = \begin{bmatrix} a_1 \ a_2 \ \vdots \ a_n \end{bmatrix}

$$

• Row matrix (vector): $\mathbf{b}$ or $\vec{b}$ $$ \mathbf{\vec{b}} = [b_1, b_2, \dots, b_m] $$

In written documents, vectors are often bolded

On chalkboards, arrows are used due to visibility constraints.

The convention in this notes will be bolded arrows for maximum readability Following both conventions at the same time for less ambiguity

🧭 Summary

Convention	Meaning / Use Case
$n \times m$	Matrix size: rows × columns
$a_{ij}$	Entry in row $i$, column $j$ of matrix $A$
$\mathbf{\vec{a}}$	Column matrix (vector)
$\mathbf{\vec{b}}$	Row matrix (vector)
Square matrix	$n = m$
Scalar matrix	$1 \times 1$