Matrices: Lecture Notes

Elements. A matrix element or entry is the content of a matrix cell (the junction of a row and a column).
- A matrix may include elements of various types (meaning, it can be heterogeneous), but usually it will contain elements of the same type (e.g., Boolean values, integers, or real numbers), and thus be homogeneous.
  - The rest of the lecture notes will focus mostly on matrices populated with numbers, since many matrix operations have significance only for numerical data.
- We refer to an element on row $i$ and column $j$ as $a_{ij}$ (
```
$a_{ij}$
```
  ), $a_{i,j}$ (
```
$a_{i,j}$
```
  ), $A(i,j)$ (
```
$A(i,j)$
```
  ), or $A[i,j]$ (
```
$A[i,j]$
```
  ).
  - In these lecture notes, we will use the notation $a_{i,j}$.
  - Examples: $a_{3,1}$, $b_{18,5}$, and $m_{6,8}$.
- Using the element notation, can also denote matrix $A$ as $A = [a_{i,j}]$.
- Note: Unlike Java, in these lecture notes, we start counting rows from $1$ and columns from $1$ (not from $0$).
  - Example: In
    $T = \begin{bmatrix} \boldsymbol{\color{green}{65.7}} & 62.4 & 73.8 & 76.6 & 75.0 \end{bmatrix}$,
    the element $65.7$ is $t_{1,1}$, not $t_{0,0}$.