Matrix Basics

Overview

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental tools in linear algebra for representing linear transformations and systems of equations.

Definition

An $m \times n$ matrix $A$ has $m$ rows and $n$ columns:

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

Entry $a_{ij}$ is in row $i$ and column $j$.

Matrix Notation

Notation	Meaning
$A = [a_{ij}]$	Matrix with entries $a_{ij}$
$A_{m \times n}$	$m \times n$ matrix
$A(i,j)$ or $a_{ij}$	Entry in row $i$, column $j$
$A_{i\cdot}$	$i$-th row of $A$
$A_{\cdot j}$	$j$-th column of $A$

Types of Matrices

Square Matrix

Number of rows equals number of columns ($n \times n$).

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Row Matrix

Single row ($1 \times n$):

$A = \begin{bmatrix} 1 & 2 & 3 & 4 \end{bmatrix}$

Column Matrix

Single column ($m \times 1$):

$A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

Zero Matrix

All entries are zero:

$O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Identity Matrix

Square matrix with 1s on main diagonal, 0s elsewhere:

$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Property: $AI = IA = A$

Diagonal Matrix

Non-zero entries only on main diagonal:

$D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$

Main Diagonal

The main diagonal consists of entries $a_{ii}$:

$\begin{bmatrix} \boxed{a_{11}} & \cdot & \cdot \\ \cdot & \boxed{a_{22}} & \cdot \\ \cdot & \cdot & \boxed{a_{33}} \end{bmatrix}$

Trace

For a square matrix $A$, the trace is the sum of diagonal entries:

$\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}$

Properties of Trace

• $\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)$
• $\text{tr}(cA) = c \cdot \text{tr}(A)$
• $\text{tr}(AB) = \text{tr}(BA)$

Matrix Equality

Two matrices $A$ and $B$ are equal if:

1. They have the same dimensions ($m \times n$)
2. All corresponding entries are equal: $a_{ij} = b_{ij}$

Examples

Example 1: Identify Matrix Type

$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

This is a:

• $3 \times 3$ matrix (square)
• Diagonal matrix
• Has trace $= 1 + 2 + 3 = 6$

Example 2: Extract Elements

For matrix:

$A = \begin{bmatrix} 2 & 5 & 7 \\ 4 & 1 & 3 \end{bmatrix}$

• $a_{12} = 5$
• $a_{23} = 3$
• First row: $\begin{bmatrix} 2 & 5 & 7 \end{bmatrix}$
• Second column: $\begin{bmatrix} 5 \\ 1 \end{bmatrix}$