Types of Matrices

3.3 Types of Matrices

Matrices come in several forms, each defined by its shape or the pattern of its entries. Different operations and properties apply to different kinds, so we examine each type, from the simplest shapes to matrices with special internal structure.

Column Matrix

A matrix with exactly one column is called a column matrix; with $m$ rows its order is $m \times 1$.

is a column matrix of order $4 \times 1$. In general notation, $A = [a_{ij}]_{m \times 1}$.

Row Matrix

A matrix with exactly one row is called a row matrix; with $n$ columns its order is $1 \times n$.

is a row matrix. In general, $B = [b_{ij}]_{1 \times n}$.

Square Matrix

A matrix is a square matrix when the number of rows equals the number of columns. If $m = n$, it is square and said to be of order $n$.

is a square matrix of order 3. In general, $A = [a_{ij}]_{m \times m}$ or simply $[a_{ij}]_{n}$.

The Diagonal of a Square Matrix

If $A = [a_{ij}]$ is square of order $n$, the elements $a_{11}, a_{22}, a_{33}, \dots, a_{nn}$ — those where the row number equals the column number — constitute the diagonal (or principal/main diagonal) of the matrix.

For

the diagonal elements are $a_{11} = 1$, $a_{22} = 4$, and $a_{33} = 6$.

Diagonal Matrix

A square matrix $B = [b_{ij}]_{m \times m}$ is a diagonal matrix if all non-diagonal elements are zero:

The diagonal elements themselves may be zero or non-zero; what matters is that every off-diagonal entry is zero.

Examples:

• Order 1: $A = [4]$
• Order 2: $B = \begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix}$
• Order 3: $C = \begin{bmatrix} -1.1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Scalar Matrix

A diagonal matrix whose diagonal elements are all equal is called a scalar matrix. That is, a square matrix $B = [b_{ij}]_{n \times n}$ with:

• $b_{ij} = 0$ when $i \neq j$ (it is diagonal), and
• $b_{ij} = k$ when $i = j$, for some constant $k$.

Examples:

• Order 1: $A = [3]$ (here $k = 3$)
• Order 2: $B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ (here $k = -1$)
• Order 3: $C = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ (here $k = 3$)

Identity Matrix

A square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0 is called an identity matrix:

The identity matrix of order $n$ is denoted $I_n$, or simply $I$ when the order is clear.

Examples:

• Order 1: $I_1 = [1]$
• Order 2: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
• Order 3: $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$