A matrix is a rectangular array of numbers arranged in rows and columns. Each element in the matrix is identified by its position, which is denoted as "aij"​, where "i" is the row number and "j" is the column number of the element.
The different types of matrices are mentioned below,
Singleton Matrix
A matrix that has only one element is called a singleton matrix. In this type of matrix number of columns and the number of rows is equal to 1. A singleton matrix is represented as [a]1⨯1.
\begin{bmatrix}5\end{bmatrix}_{1 \times 1}
Null Matrix
A matrix whose all elements are zero is called a Null Matrix. A null matrix is also called a Zero Matrix because all its elements are zero.
\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix}_{3 \times 3}
Row Matrix
A matrix that contains only one row and any number of columns is known as a row matrix. A row matrix is represented as [a]1⨯n, where 1 is the number of rows and n is the number of columns present in a row matrix.
\begin{bmatrix}1 & 3 & 7 \\\end{bmatrix}_{1 \times 3}
Column Matrix
A matrix that contains only one column and any number of rows is called a Column Matrix. A Column Matrix is represented as [a]n⨯1 where n is the number of rows and 1 is the number of columns.
\begin{bmatrix}1 \\14 \\4 \\5\end{bmatrix}_{4 \times 1}
Horizontal Matrix
A matrix in which the number of rows is lower than the number of columns is called a Horizontal Matrix.
\begin{bmatrix}1 & 2 & 3 & 4 \\5 & 6 & 7 & 8\end{bmatrix}_{2 \times 4}
Vertical Matrix
The matrix in which the number of rows exceeds the number of columns is called a Vertical Matrix. A Vertical matrix is represented as [a]i⨯j where i > j.
\begin{bmatrix}1 & 2 \\3 & 4 \\5 & 6 \\7 & 8\end{bmatrix}_{4 \times 2}
Rectangular Matrix
A matrix that does not have an equal number of rows and columns is known as a Rectangular Matrix. A rectangular matrix can be represented as [A]m×n where m ≠n.
\begin{bmatrix}1 & 3 & 7 & 15 \\3 & 4 & 6 & 11 \\5 & 2 & 9 & 8\end{bmatrix}_{3 \times 4}
In the above example, we see that the number of rows is 3 while the number of columns is 4, i.e. both are unequal, thus making it a rectangular matrix. We can say that both horizontal and vertical matrices are examples of rectangular matrices.
Square Matrix
A matrix that has an equal number of rows and an equal number of columns is called a Square Matrix. Generally, the representation used for the square matrix is [A]n×n.
\begin{bmatrix}8 & 3 & 2 \\6 & 4 & 6 \\5 & 7 & 9\end{bmatrix}_{3 \times 3}
Diagonal Matrix
A matrix that has all elements as 0 except diagonal elements is known as a diagonal matrix. A Diagonal Matrix is only possible in the case of a Square Matrix.
\begin{bmatrix}8 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 9\end{bmatrix}_{3 \times 3}
Scalar MatrixÂ
A diagonal matrix whose all diagonal elements are non-zero and the same is called a Scalar Matrix. Scalar Matrix is a kind of diagonal matrix where all diagonal elements are the same. Identity Matrix is a special case of Scalar Matrix.
\begin{bmatrix}4 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 4\end{bmatrix}_{3 \times 3}
Identity Matrix
A diagonal matrix where all the diagonal elements are 1 and all non-diagonal elements are 0 is called an Identity Matrix. The Identity Matrix is called the Unit Matrix. The identity matrix or unit matrix always has an equal number of rows and columns.
\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}_{3 \times 3}
Triangular Matrix
A square matrix in which the non-zero elements form a triangular below and above the diagonal is called a Triangular Matrix. Based on the triangle formed below or above the diagonal, the triangular matrix is classified as:
- Upper Triangular Matrix
- Lower Triangular Matrix
Upper Triangular Matrix
A square matrix in which all the elements below the diagonal are zero and the elements from the diagonal and above are non-zero elements is called an Upper Triangular Matrix. In an Upper Triangular Matrix, the non-zero elements form a triangular-like shape.
\begin{bmatrix}8 & 5 & 6 \\0 & 4 & 7 \\0 & 0 & 9\end{bmatrix}_{3 \times 3}
Lower Triangular Matrix
A square matrix in which all the elements above the diagonal are zero and the elements from the diagonal and below are non-zero elements is called a Lower Triangular Matrix. In a Lower Triangular Matrix, the non-zero elements form a triangular-like shape from the diagonal and below.
\begin{bmatrix}8 & 0 & 0 \\6 & 4 & 0 \\5 & 7 & 9\end{bmatrix}_{3 \times 3}
Singular Matrix
A singular matrix is referred to as a square matrix whose determinant is zero and is not invertible.
If det A = 0, a square matrix "A" is said to be singular; otherwise, it is said to be non-singular.
A = \begin{bmatrix}3 & 6 & 9 \\6 & 12 & 18 \\2 & 4 & 6\end{bmatrix}
⇒ |A| = 3(12 × 6 - 18 × 4) - 6(6 × 6 - 18 × 2) + 9(6 × 4 - 12 × 2)
⇒ |A| = 3(72 - 72) - 6(36 - 36) + 9(24 - 24)
⇒ |A| = 3 × 0 - 6 × 0 + 9 × 0 = 0
Non Singular Matrix
A Non-Singular matrix is defined as a square matrix whose determinant is not equal to zero and is invertible.
|A| =\begin{bmatrix}1 & 5 \\9 & 8\end{bmatrix} ⇒ |A| = 8 × 1 - 9 × 5 = 8 - 45 = -37
Symmetric Matrix
A square matrix "A" of any order is defined as a symmetric matrix if the transpose of the matrix is equal to the original matrix itself, i.e., AT = A.
|A| =\begin{bmatrix}2 & 1 \\1 & 2\end{bmatrix}
Skew Symmetric Matrix
A square matrix "A" of any order is defined as a skew-symmetric matrix if the transpose of the matrix is equal to the negative of the original matrix itself, i.e., AT = -A.
\begin{bmatrix}0 & 3 & 5 \\-3 & 0 & -2 \\-5 & 2 & 0\end{bmatrix}
Orthogonal Matrix
A square matrix whose transpose is equal to its inverse is called Orthogonal Matrix. In an Orthogonal Matrix if AT = A-1 then AAT = I where I is the Identity Matrix.
A =\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\\sin(\theta) & \cos(\theta)\end{bmatrix} andA^{T} =\begin{bmatrix}\cos(\theta) & \sin(\theta) \\-\sin(\theta) & \cos(\theta)\end{bmatrix}
A \times A^{T} =\begin{bmatrix}\cos^{2}(\theta) + \sin^{2}(\theta) &\cos(\theta)\sin(\theta) - \cos(\theta)\sin(\theta)\\[6pt]\sin(\theta)\cos(\theta) - \cos(\theta)\sin(\theta) &\cos^{2}(\theta) + \sin^{2}(\theta)\end{bmatrix}
\Rightarrow\ A \times A^{T} =\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}= I_{(2 \times 2)}
Idempotent Matrix
An idempotent matrix is a special type of square matrix that remains unchanged when multiplied by itself, i.e., A2 = A.
A =\begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix} Hence,
A \cdot A =\begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix}\cdot\begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix}=\begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix}= A
Nilpotent Matrix
A Nilpotent is a square matrix that when raised to some positive power results in a zero matrix. The least power let's say 'p' for which the matrix yields zero matrices, then it is called the Nilpotent Matrix of power 'p'.
A =\begin{bmatrix}0 & 1 & 2 \\0 & 0 & 3 \\0 & 0 & 0\end{bmatrix}
\Rightarrow A^{2} =\begin{bmatrix}0 & 1 & 2 \\0 & 0 & 3 \\0 & 0 & 0\end{bmatrix}\cdot\begin{bmatrix}0 & 1 & 2 \\0 & 0 & 3 \\0 & 0 & 0\end {bmatrix}
=\begin{bmatrix}0 & 0 & 3 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix}
\Rightarrow A^{2} =\begin{bmatrix}0 & 0 & 3 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix} and A3 = A . A2
\Rightarrow A^{3} =\begin{bmatrix}0 & 1 & 2 \\0 & 0 & 3 \\0 & 0 & 0\end{bmatrix}\cdot\begin{bmatrix}0 & 0 & 3 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix}=\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix} Hence, A is a Nilpotent Matrix of index 3.
Periodic Matrix
A periodic matrix is a square matrix that exhibits periodicity, meaning there exists a positive integer p such that when the matrix is raised to the power p+1, it equals the original matrix (Ap+1 = A). If p = 1 then A2 = A it means A is an Idempotent Matrix. Thus we can say that the Idempotent Matrix is a case of the Periodic Matrix.
A = \begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix}
The above square matrix is a Periodic Matrix of Period 2, where p = 1.
Involutory Matrix
An involutory matrix is a special type of square matrix whose inverse is the original matrix itself, i.e., P = P-1, or, in other words, its square is equal to an identity matrix i.e. P2 = I.
A = \begin{bmatrix}2 & 1 \\-3 & -2\end{bmatrix}
Hermitian Matrix
A complex square matrix is called a Hermitian Matrix if the conjugate transpose of the matrix is equal to the original matrix. In this type of matrix, the diagonal elements must be a real number.
A = \begin{bmatrix}2 & 1 \\-3 & -2\end{bmatrix}
Skew Hermitian Matrix
A complex square matrix is called a Skew Hermitian Matrix if the conjugate transpose of the matrix is equal to the negative of the original matrix. In this type of matrix, the diagonal elements can be either 0 or a complex number but can not be real numbers other than 0.
A = \begin{bmatrix}2i & 2i & -3i \\-2i & 0 & 4 \\3i & -4 & 0\end{bmatrix}
Boolean Matrix
The matrix which represents the binary relationship and takes 0 and 1 as its element is called a Boolean Matrix.
\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 1 & 0\end{bmatrix}
Stochastic Matrix
A square matrix represents probability data, meaning that each element is non-negative, and the sum of the elements in each row is equal to 1. Such a matrix is called a stochastic matrix.
\begin{bmatrix}0.2 & 0.5 & 0.3 \\0.1 & 0.3 & 0.6 \\0.4 & 0.2 & 0.4\end{bmatrix}
Summary
Different types of matrices in linear algebra along with their representation is given below.
| Matrix Type | Example |
|---|---|
| Singleton Matrix | |
| Null Matrix | |
| Row Matrix | |
| Column Matrix | |
| Horizontal Matrix | |
| Vertical Matrix | |
| Rectangular Matrix | |
| Square Matrix | |
| Diagonal Matrix | |
| Scalar Matrix | |
| Identity Matrix | |
| Triangular Matrix | |
| Singular Matrix | |A| = 0; determinant is zero |
| Non-Singular Matrix | |A| ≠0; determinant is not equal to zero |
| Symmetric Matrix | AT = A; square matrix that remains same when its transpose is taken |
| Skew-Symmetric Matrix | AT = -A; square matrix with transpose equal to negative |
| Orthogonal Matrix | Q QT = ln = QT Q |
| Idempotent Matrix | A*A = A |
| Nilpotent Matrix | Ak = 0; where, k ≤ n |
| Periodic Matrix | A(k+1) = A |
| Involutory Matrix | A*A = I |
| Hermitian Matrix | |
| Skew Hermitian Matrix | If A is a skew-hermitian matrix, then A* = -A |
| Boolean Matrix | |
| Stochastic Matrix |
