PX153 - K1 - terminology of matrices

introduction

a matrix is a rectangular array of numbers
if there are $m$ rows and $n$ columns, the matrix has a size of $m \times n :$

A_{m \times n} = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & ⋮ \\ ⋮ & ⋱ & ⋮ \\ a_{m 1} & . . . & . . . & a_{m n} \end{matrix})

$a_{i j}$ represents the element in the $i^{t h}$ row and $j^{t h}$ column
often written as:

A = (a_{i j})_{m \times n}

if a matrix has shape, $m \times 1$ , it is a column vector:

A = (\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{m} \end{matrix})

if a matrix has shape, $1 \times n$ , it is a row vector:

B = (\begin{matrix} b_{1} & b_{2} & \dots & b_{n} \end{matrix})

transpose

if a matrix, $A = (a_{i j})_{m \times n}$ , the transpose of $A$ is the matrix, $A^{T}$ , has dimensions $n \times m$ , and is obtained by transposing the rows and columns of $A :$

A^{T} = (α_{i j})_{n \times m} = (a_{j i})_{n \times m}

eg:

\begin{aligned} A & = (\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}) \\ A^{T} & = (\begin{array}{c} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}) \end{aligned}

special matrices

$m \times n$ matrices with all elements equal to $0$ is denoted by ' $0_{m \times n}$ ', and is called a zero matrix
a $n \times n$ matrix is called a square matrix of order $n$
- $a_{i j}$ , for $i = j$ , are called the elements of the diagonal
a square matrix is a diagonal matrix if $a_{i j} = 0 \forall i \neq j$
a diagonal matrix is an identity matrix if the diagonal elements are $1$