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:

// 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