# Definition and Types

### Definition of Matrices

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In general, a matrix with m rows and n columns is called an $$m × n$$ matrix.

#### General form of a matrix A:

#### $$A = \begin{bmatrix} a\_{11} & a\_{12} & \dots & a\_{1n} \ a\_{21} & a\_{22} & \dots & a\_{2n} \ \vdots & \vdots & \ddots & \vdots \ a\_{m1} & a\_{m2} & \dots & a\_{mn} \end{bmatrix}$$

Where $$a\_{ij}$$ represents the element in the i-th row and j-th column.

In PHP matrices can be written like n-dimensional arrays:

```php
// 1-dimensional array
$matrixD1 = [1, 2];

// 2-dimensional array
$matrixD2 = [
    [1, 2],[3, 4],[5, 6]
];

// 3-dimensional array
$matrixD3 = [
    [
        [1, 2, 3],
        [4, 5, 6],
        [7, 8, 9]
    ],
    [
        [10, 11, 12],
        [13, 14, 15],
        [16, 17, 18]
    ],
    [
        [19, 20, 21],
        [22, 23, 24],
        [25, 26, 27]
    ]
];
```

### Types of Matrices

#### Row matrix:

A matrix with only one row $$(1 × n)$$.

Example: $$\begin{bmatrix} 1\ 2\ 3\ 4 \end{bmatrix}$$

#### Column matrix:

A matrix with only one column $$(m × 1)$$.

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

#### Square matrix:

A matrix with an equal number of rows and columns $$(n × n)$$.

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

#### Diagonal matrix:

A square matrix where all elements outside the main diagonal are zero.

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

#### Identity matrix:

A square matrix with 1s on the main diagonal and 0s elsewhere, usually denoted by $$I$$.

Example: $$A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$$

#### Zero matrix:

A matrix where all elements are zero, usually denoted by 0.

Example: $$A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$$

#### Symmetric matrix:

A square matrix that is equal to its transpose $$(A = A^T)$$.

Example: $$A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \end{bmatrix}$$

#### Skew-symmetric matrix:

A square matrix A where .

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

#### Upper triangular matrix:

A square matrix where all elements below the main diagonal are zero.

Example: $$A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}$$

#### Lower triangular matrix:

A square matrix where all elements above the main diagonal are zero.

Example: $$A = \begin{bmatrix} 1 & 0 & 0 \ 2 & 3 & 0 \ 4 & 5 & 6 \end{bmatrix}$$

Understanding these types of matrices is crucial for working with more advanced concepts in linear algebra, such as matrix operations, determinants, and eigenvalues