Types of Matrices

A Matrix in the plural (Matrices) is a rectangular table of numbers, symbols, or expressions arranged in particular rows and columns. It is used to represent a mathematical expression or a property of such an expression.

In Mathematics, Matrices are also used for representing geometric transformations like rotations or coordinating changes. Many mathematical problems are reduced by using them to a matrix computation, which often involves computing with matrices of enormous dimension. Matrices are used in many areas of mathematics and most scientific fields, either directly or through their use in geometry and numerical analysis.

What is a Matrix?

A matrix is a rectangular table of numbers, which are called the entries of the matrix. Matrices are subjected to simple operations, such as addition subtraction. A real matrix and a complex matrix are matrices whose entries are respectively real and complex numbers.

Size of a matrix: The size of a matrix is only determined by the number of rows and columns it has. There is no limit to the numbers of rows and columns of a matrix as long as they are positive integers in them. A matrix with rows and b columns is called a × b matrix, or a-by-b matrix, while m and b are called its dimensions.

Types of Matrices:

1. Row Matrix

This is a type of matrix that only contains one row and any number of columns is known as a row matrix. 

2. Column Matrix

These are the types of matrices that contain only one column and any number of rows is known as a column matrix. 

3. Singleton Matrix

These are the matrix types, which have only one element, known as singleton matrix. In this type of matrix, the number of columns and the number of rows equals 1.

4. 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

5. Square Matrix

These are the types of matrices with an equal number of rows and an equal number of columns known as a square matrix. Generally, the representation used for the square matrix is [A].

6. Null Matrix

These are the types of matrices, which have all elements as 0, known as the null matrix. 

7. Diagonal Matrix

These are the types of matrices with all elements as 0 except diagonal elements and are known as diagonal matrices.

8. Scalar Matrix 

These are the types of matrices that have all elements as 0 except diagonal elements and all diagonal elements are the same is, known as a scalar matrix. It is a kind of diagonal matrix where all diagonal elements are the same.

9. Identity Matrix

It is a kind of scalar matrix where all the diagonal elements are 1 and all non-diagonal elements are 0. The identity matrix always has an equal number of rows and columns. 

10. Upper Triangular Matrix

These are the types of matrix, which has a kind of square matrix that has all elements as 0 below the diagonal.

11. Lower Triangular Matrix

These are the types of matrix, which has a kind of square matrix in which all the elements above the diagonal are 0.

12. Singular Matrix 

The type of matrix where the value of the determinant is equal to 0.

13. Non-singular Matrix

The types of a matrix where the value of the determinant is not equal to 0.

14. Symmetric matrix

A matrix is called a Symmetric Matrix, when a[column x row]=a[row x column]. 

15. Skew-Symmetric Matrix

A matrix is called a Skew Symmetric Matrix when all values of det A are equal to 0.

Some Important Points Regarding Skew Symmetric and Symmetric Matrix:-

Suppose, A is a square matrix, then A + A’ is a symmetric matrix. A – A’ becomes a skew-symmetric matrix.

1. All Square Matrices are uniquely used in expressions as the sum of a symmetric matrix and a skew-symmetric matrix.
2. Suppose, A and B are symmetric matrices, then AB is a symmetric matrix. So, AB = BA,
3. The matrix B AB is a symmetric matrix or skew-symmetric matrix; if A is an asymmetric or skew-symmetric matrix.
4. All positive integral powers of a symmetric matrix are symmetric in nature.
5. All the Positive odd integrals are powers of a skew-symmetric matrix. They are skew-symmetric and positive; even integral powers of a skew-symmetric matrix are symmetric.

Conclusion

Matrices represent the mathematical linear maps and thus allow explicit computations in linear expressions. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. Not all matrices are related to linear algebra. The case in graph theory of incidence, and adjacency matrix. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.