What is Matrix? | Types of Matrix

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

  1. Row matrix
  2. Column matrix
  3. Null matrix
  4. Square matrix
  5. Identity matrix or Unit matrix
  6. Diagonal matrix
  7. Scalar matrix
  8. Singular matrix
  9. Non-singular matrix
  10. Upper triangular matrix
  11. Lower triangular matrix
  12. Idempotent matrix
  13. Involutory matrix
  14. Nilpotent matrix
  15. Real matrix
  16. 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.

  1. Symmetric matrix
  2. Skew symmetric matrix
  3. 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.

  1. Hermitian matrix
  2. Skew hermitian matrix
  3. 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.

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