Types of matrices

Following are the main Types of Matrices

1. Square Matrix

Any matrix whose number of rows will equal to the number of column, that matrix is called square matrix. It will look like square shape. 

For example 

 number of rows = no. of columns 

3 = 3

so, this matrix is square matrix. 

2. Rectangular Matrix

If any matrix whose number of rows are not equal the number of columns, that matrix is called rectangular matrix. Its shape looks like rectangular. 

for example

no. of rows are = 3

no. of columns are = 2

its number of rows is not equal to no. of column, so this matrix is called rectangular matrix.

3. Diagonal Matrix 

Diagonal matrix is square matrix whose all elements are zero except main diagonals are not zero. 

for example

Above matrix's number of rows and number of columns are 3 and Its diagonal starts from left and close at right which are not zero which are 2.5 and 7

4. Scalar Matrix 

Scalar matrix is diagonal matrix whose main diagonal are equal. It means, it has 3 condition

1 It must be square matrix 

2 it must be diagonal matrix

3  it must be equal main diagonal 

For Example 

Above matrix is diagonal matrix and its main diagonal elements are 2,2 and 2 which are equal. So, it is scalar matrix. 

5. Identity or unit matrix

Unit or identity matrix is diagonal matrix whose main diagonal must be 1

6. Nil or zero matrix

Any matrix whose all elements are zero that matrix is called nil or zero matrix

7. Row matrix

Any matrix is called row matrix  who has one row and it may be 1 column or more than one column. 

8. Column matrix

Any matrix is called column matrix who has only one column but it may have one or more rows. 

9. Transpose matrix

If we change the rows of matrix into columns or same matrix and columns change into its row. After this, a new matrix is called transpose of matrix. 

10.Symmetric matrix

Symmetric matrix is the transpose of matrix if it is equal to its original matrix. 

11. Skew symmetric matrix

Skew Symmetric matrix is the transpose of matrix if it is equal to the negative of its original matrix. It means all the elements of the transpose of matrix must be same but negative. 

12. Sub matrix

If we withdraw some rows and some columns of original matrix and make the new matrix. This new matrix is called sub matrix. 

13. Equal matrix

Two or more matrix is equal if 

1. Its all elements must be same 

2. and its number of columns and number of rows must be equal. 

14. Upper Triangular matrix

Upper triangular matrix is the square matrix who make the triangle right side with its elements and all other elements must be zero.

15. Lower Triangular matrix

Lower  triangular matrix is the square matrix who make the triangle left side with its elements and all other elements must be zero.

Tough types of Matrix

16. Conjugate of a Matrix

Matrix obtain from any given matrix A, after replacing its elements by complex conjugates, is called the conjugate of A and is generally dnoted A_

 

17. hermitian and Skew Hermitian Matrix

A square matrix A is said to be hermitian if every diagonal elements is real 

and aij = aji

[ 5 5+6i ]

[ 5-6i 3 ]

Skew hermitian matrix

If square matrix A = aij is said to be skew hermitain if aij = aji , the elements is the negative conjugate of 9 j,i) the elements 

[ 5         5+6i ]

[ -5-6i        3 ]

18. Idempotent Matrix 

A matrix is said to be idempotent when A^2 = A

19.  Involutory Matrix 

A matrix A is said to be involutory when involutory matrix must equal  its multiplication of equal matrix to its transpose 

A^2 = I 

20. Nilpotent Matrix

If n is the least positive integer for which A^n = O , then A is called nilpotent matrix, and n is called the index of the matrix. 

21. Orthogonal Matrix

A square matrix A with real elements is said to be orthogonal matrix. 

22. Unitary Matrix 

A square matrix A is said to be unitary if A^dalta X A = I