03 - Matrix

Matrix

Two-dimensional array of numbers

$m\times n$ matrix

Array of numbers arranged in $m$ rows and $n$ columns

Entries of a $m\times n$ matrix $A$

Generally

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)$$

Entry in the $i$th row and $j$th column
→ Notation: $a_{ij}$ ($1\le i\le m$ and $1\le j\le n$)

Then we can write

$$A=(a_{ij})$$

Rows and Columns of matrix $A$

$$i\text{th row}=(a_{i_1},a_{i_2},\cdots ,a_{in})$$ $$j\text{th column}=\left(\begin{array}{c}{a}_{1j}\\ a_{2j}\\ ⋮\\ a_{mj}\end{array}\right)$$

Matrix Notation

• $M_{mn}$: Set of real $m\times n$ matrices
• $M_{1n}={\mathbb{R}}^n$: Set of real $n-$row vectors
• $M_{n_1}={\mathbb{R}}_{\mathrm{col}}^n$: Set of real $n$-column vectors