Matrix Types

Row Matrix

A row matrix is a matrix that consists of only one row.

The order of a row matrix is $1 \times n$ , where $n$ is the number of columns.

Its general form is:

A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{bmatrix}

Example:

P = \begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix}

Matrix $P$ is a row matrix of order $1 \times 4$ .

Column Matrix

A column matrix is a matrix that consists of only one column.

The order of a column matrix is $m \times 1$ , where $m$ is the number of rows.

Its general form is:

B_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}

Example:

Q = \begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix}

Matrix $Q$ is a column matrix of order $3 \times 1$ .

Square Matrix

A square matrix is a matrix that has the same number of rows and columns.

If $\text{number of rows} = \text{number of columns} = n$ , then the matrix is of order $n \times n$ .

Its general form is:

A_{n \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix}

In a square matrix, there are:

Main Diagonal (or Principal Diagonal):

The elements $a_{11}, a_{22}, \dots, a_{nn}$ (i.e., $a_{ij}$ where $i=j$ ).
Anti-diagonal (or Counter-diagonal):

The elements $a_{1n}, a_{2,n-1}, \dots, a_{n1}$ (i.e., $a_{ij}$ where $i+j=n+1$ ).

Example:

M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Matrix $M$ is a square matrix of order $3 \times 3$ . Its main diagonal elements are $1, 5, 9$ . Its anti-diagonal elements are $3, 5, 7$ .

Rectangular Matrix

A rectangular matrix is a matrix where the number of rows and columns are not equal ( $m \neq n$ ).

General example:

C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

Matrix $C$ above has $2$ rows and $3$ columns, so its order is $2 \times 3$ . Since the number of rows is not equal to the number of columns ( $2 \neq 3$ ), matrix $C$ is a rectangular matrix.

Rectangular matrices with more columns than rows are often called wide, while those with more rows than columns are often called tall.

Wide Matrix

A wide matrix is a rectangular matrix with more columns than rows ( $n > m$ ).

Example:

D = \begin{bmatrix} 1 & 0 & 4 \\ -2 & 3 & 5 \end{bmatrix}

Matrix $D$ is a wide matrix of order $2 \times 3$ .

Tall Matrix

A tall matrix is a rectangular matrix with more rows than columns ( $m > n$ ).

Example:

T = \begin{bmatrix} 7 & 1 \\ 0 & -3 \\ 4 & 2 \end{bmatrix}

Matrix $T$ is a tall matrix of order $3 \times 2$ .

A triangular matrix is a square matrix where the elements on one side of the main diagonal are all zero. The diagonal entries themselves may be anything. Upper and lower triangular matrices differ in which side is zero.

Upper Triangular Matrix

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

In symbols, $a_{ij} = 0$ whenever $i > j$ .

Example:

U = \begin{bmatrix} 5 & 2 & -1 \\ 0 & 3 & 7 \\ 0 & 0 & 1 \end{bmatrix}

Lower Triangular Matrix

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

In symbols, $a_{ij} = 0$ whenever $i < j$ .

Example:

L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 1 & 9 \end{bmatrix}

Diagonal Matrix

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

In symbols, $a_{ij} = 0$ whenever $i \neq j$ . Elements on the main diagonal can be zero or non-zero.

Example:

X = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Matrix $X$ is a diagonal matrix of order $3 \times 3$ .

Identity Matrix

An identity matrix (denoted by $I$ or $I_n$ ) is a diagonal matrix where all elements on the main diagonal are $1$ .

Example:

I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The identity matrix is the neutral element of matrix multiplication. If $A$ has dimensions $m \times n$ , the identity matrix must match the side on which it is multiplied:

I_mA=A=AI_n

Zero Matrix

A zero matrix (denoted by $O$ or $O_{m \times n}$ ) is a matrix where all elements are zero.

Example:

O_{2 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Symmetric Matrix

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

For every pair of indices $i$ and $j$ , a symmetric matrix satisfies $a_{ij} = a_{ji}$ . Entries on opposite sides of the main diagonal are equal.

Example:

S = \begin{bmatrix} 1 & 7 & -3 \\ 7 & 2 & 0 \\ -3 & 0 & 5 \end{bmatrix}

In matrix $S$ :

$s_{12} = s_{21} = 7$
$s_{13} = s_{31} = -3$
$s_{23} = s_{32} = 0$

Row Matrix

A row matrix is a matrix that consists of only one row.

The order of a row matrix is $1 \times n$ , where $n$ is the number of columns.

Its general form is:

A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{bmatrix}

Example:

P = \begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix}

Matrix $P$ is a row matrix of order $1 \times 4$ .

Column Matrix

A column matrix is a matrix that consists of only one column.

The order of a column matrix is $m \times 1$ , where $m$ is the number of rows.

Its general form is:

B_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}

Example:

Q = \begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix}

Matrix $Q$ is a column matrix of order $3 \times 1$ .

Square Matrix

A square matrix is a matrix that has the same number of rows and columns.

If $\text{number of rows} = \text{number of columns} = n$ , then the matrix is of order $n \times n$ .

Its general form is:

A_{n \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix}

In a square matrix, there are:

Main Diagonal (or Principal Diagonal):

The elements $a_{11}, a_{22}, \dots, a_{nn}$ (i.e., $a_{ij}$ where $i=j$ ).
Anti-diagonal (or Counter-diagonal):

The elements $a_{1n}, a_{2,n-1}, \dots, a_{n1}$ (i.e., $a_{ij}$ where $i+j=n+1$ ).

Example:

M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Matrix $M$ is a square matrix of order $3 \times 3$ . Its main diagonal elements are $1, 5, 9$ . Its anti-diagonal elements are $3, 5, 7$ .

Rectangular Matrix

A rectangular matrix is a matrix where the number of rows and columns are not equal ( $m \neq n$ ).

General example:

C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

Matrix $C$ above has $2$ rows and $3$ columns, so its order is $2 \times 3$ . Since the number of rows is not equal to the number of columns ( $2 \neq 3$ ), matrix $C$ is a rectangular matrix.

Rectangular matrices with more columns than rows are often called wide, while those with more rows than columns are often called tall.

Wide Matrix

A wide matrix is a rectangular matrix with more columns than rows ( $n > m$ ).

Example:

D = \begin{bmatrix} 1 & 0 & 4 \\ -2 & 3 & 5 \end{bmatrix}

Matrix $D$ is a wide matrix of order $2 \times 3$ .

Tall Matrix

A tall matrix is a rectangular matrix with more rows than columns ( $m > n$ ).

Example:

T = \begin{bmatrix} 7 & 1 \\ 0 & -3 \\ 4 & 2 \end{bmatrix}

Matrix $T$ is a tall matrix of order $3 \times 2$ .

Triangular Matrix

Upper Triangular Matrix

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

In symbols, $a_{ij} = 0$ whenever $i > j$ .

Example:

U = \begin{bmatrix} 5 & 2 & -1 \\ 0 & 3 & 7 \\ 0 & 0 & 1 \end{bmatrix}

Lower Triangular Matrix

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

In symbols, $a_{ij} = 0$ whenever $i < j$ .

Example:

L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 1 & 9 \end{bmatrix}

Diagonal Matrix

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

In symbols, $a_{ij} = 0$ whenever $i \neq j$ . Elements on the main diagonal can be zero or non-zero.

Example:

X = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Matrix $X$ is a diagonal matrix of order $3 \times 3$ .

Identity Matrix

An identity matrix (denoted by $I$ or $I_n$ ) is a diagonal matrix where all elements on the main diagonal are $1$ .

Example:

I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The identity matrix is the neutral element of matrix multiplication. If $A$ has dimensions $m \times n$ , the identity matrix must match the side on which it is multiplied:

I_mA=A=AI_n

Zero Matrix

A zero matrix (denoted by $O$ or $O_{m \times n}$ ) is a matrix where all elements are zero.

Example:

O_{2 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Symmetric Matrix

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

For every pair of indices $i$ and $j$ , a symmetric matrix satisfies $a_{ij} = a_{ji}$ . Entries on opposite sides of the main diagonal are equal.

Example:

S = \begin{bmatrix} 1 & 7 & -3 \\ 7 & 2 & 0 \\ -3 & 0 & 5 \end{bmatrix}

In matrix $S$ :

$s_{12} = s_{21} = 7$
$s_{13} = s_{31} = -3$
$s_{23} = s_{32} = 0$