A rectangular arrangement of numbers into rows and columns is known as a matrix. The horizontal lines are called rows and the vertical lines are called columns. The matrix is enclosed by [] or ().
Let A be any matrix. It can be written as A = [aij]m × n
\[A=[a_{ij}]_{m\times n}=\begin{bmatrix} a_{11} & a_{12} & … & … & a_{1n}\\ a_{21} & a_{22} & … & … & a_{2n}\\ … & … & … & … & …\\ … & … & … & … & …\\ a_{m1} & a_{m2} & … & … & a_{mn} \end{bmatrix}\]
where, aij represents the element of row ‘i’ and column ‘j’. m × n represents the order of the matrix. It states that there are m rows and n columns in matrix A. Here i = 1, 2, 3, ……, m and j = 1, 2, 3, ……, n.
The plural of matrix is matrices. When we talk about more than one matrix then we called matrices.
Types of Matrix
The special types of matrix are
- Row matrix
- Column matrix
- Null matrix
- Square matrix
- Identity matrix or Unit matrix
- Diagonal matrix
- Scalar matrix
- Singular matrix
- Non-singular matrix
- Upper triangular matrix
- Lower triangular matrix
- Idempotent matrix
- Involutory matrix
- Nilpotent matrix
- Real matrix
- Complex matrix
1. Row matrix
A matrix having only one row is called row matrix. For example,
\[\begin{bmatrix} 9 & 0 & -0.35 & 12 & 8 \end{bmatrix}\]
2. Column matrix
A matrix having only one column is called column matrix. For example,
\[\begin{bmatrix} 8\\ 1\\ 2\\ 0 \end{bmatrix}\]
3. Null matrix
A matrix that has all of its elements equal to zero is called null matrix. It is also known as zero matrix.
\[\begin{bmatrix} 0\\ 0\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}\]
You can read more about null matrix here.
4. Square matrix
A matrix that has equal number of rows and columns is called square matrix. For example,
\[\begin{bmatrix} 3 \end{bmatrix}, \begin{bmatrix} 2 & 3\\ 4 & 5 \end{bmatrix}, \begin{bmatrix} 4 & 6 & 7\\ 0 & 4 & 8\\ -4 & -9 & 0 \end{bmatrix}\]
You can read more about square matrix here.
5. Identity matrix
A square matrix that has all of its diagonal elements equal to one and all of its off-diagonal elements equal to zero, is called identity matrix. It is also known as unit matrix. For example,
\[\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\]
You can read more about identity matrix or unit matrix here.
6. Diagonal matrix
A square matrix that has all of its off-diagonal elements equal to zero, is called diagonal matrix. For example,
\[\begin{bmatrix} 1 & 0\\ 0 &9 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & -2 \end{bmatrix}\]
You can read more about diagonal matrix here.
7. Scalar matrix
A square matrix that has all of its diagonal elements equal and all of its off-diagonal elements equal to zero, is called scalar matrix. It is a special case of diagonal matrix in which all diagonal elements are equal. For example,
\[\begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \begin{bmatrix} -5 & 0 & 0\\ 0 & -5 & 0\\ 0 & 0 & -5 \end{bmatrix}\]
You can read more about scalar matrix here.
8. Singular matrix
A square matrix whose determinant is equal to zero, is called singular matrix. For example,
\[\begin{bmatrix} 1 &2 \\ 3& 6 \end{bmatrix}, \begin{bmatrix} 3 &8 &1 \\ -4& 1 &1 \\ -4&1 & 1 \end{bmatrix}\]
You can read more about singular matrix here.
9. Non-Singular matrix
A square matrix whose determinant is not equal to zero, is called non-singular matrix. For example,
\[\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 1 &2 &3 \\ 4& 5 &6 \\ 7&8 & 9 \end{bmatrix}\]
You can read more about non-singular matrix here.
10. Upper triangular matrix
A square matrix whose lower off-diagonal elements are equal to zero, is called upper triangular matrix. For example,
\[\begin{bmatrix} 1 &2 \\ 0& -4 \end{bmatrix}, \begin{bmatrix} 1 &2 &3 \\ 0& 5 &6 \\ 0&0 & 9 \end{bmatrix}\]
You can read more about upper triangular matrix here.
11. Lower triangular matrix
A square matrix whose upper off-diagonal elements are equal to zero, is called lower triangular matrix. For example,
\[\begin{bmatrix} 1 &0 \\ 3& 6 \end{bmatrix}, \begin{bmatrix} 1 &0 &0 \\ 8& 5 &0 \\ 2&-1 & 9 \end{bmatrix}\]
You can read more about lower triangular matrix here.
12. Idempotent matrix
A square matrix P is said to be idempotent matrix if P2 = P. For example,
\[\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}, \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 2 &-2 &-4 \\ -1& 3 &4 \\ 1&-2 & -3 \end{bmatrix}\]
You can read more about idempotent matrix here.
13. Involutory matrix
A square matrix P is said to be involutory matrix if P2 = I, where I is the identity matrix. For example,
\[\begin{bmatrix} 4 &-1 \\ 15& -4 \end{bmatrix}, \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 4 &3 &3 \\ -1& 0 &-1 \\ -4&-4 & -3 \end{bmatrix}\]
You can read more about involutory matrix here.
14. Nilpotent matrix
A square matrix P is said to be nilpotent matrix of class z if Pz = O, where O is the null matrix. For example,
\[B=\begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}\]
It is a nilpotent matrix of class 2 i.e. B2 = O
\[D=\begin{bmatrix} 1 &1 &3 \\ 5& 2 &6 \\ -2&-1 & -3 \end{bmatrix}\]
It is a nilpotent matrix of class 3 i.e. D3 = O
You can read more about nilpotent matrix here.
15. Real matrix
A matrix that has all elements taking real values is called a real matrix.
Based on the relationship between matrix and its transpose, a real matrix can be classified into three types i.e.
- Symmetric matrix
- Skew symmetric matrix
- Orthogonal matrix
a. Symmetric matrix
A square matrix P is said to be symmetric matrix if PT = P.
You can read more about symmetric matrix here.
b. Skew symmetric matrix
A square matrix P is said to be skew-symmetric matrix if PT = -P.
You can read more about skew symmetric matrix here.
c. Orthogonal matrix
A square matrix P is said to be orthogonal matrix if PT = P-1.
You can read more about orthogonal matrix here.
16. Complex matrix
A matrix that has atleast one element taking complex value is called a complex matrix.
Based on the relationship between matrix and its conjugate transpose, a complex matrix can be classified into three types i.e.
- Hermitian matrix
- Skew hermitian matrix
- Unitary matrix
a. Hermitian Matrix
A complex square matrix P is said to be hermitian matrix if Pθ = P.
You can read more about hermitian matrix here.
b. Skew hermitian matrix
A complex square matrix P is said to be skew-hermitian matrix if Pθ = -P.
You can read more about skew hermitian matrix here.
c. Unitary matrix
A complex square matrix P is said to be unitary matrix if Pθ = P-1.
You can read more about unitary matrix here.
