Types of Matrix

Matrix is one of the most important topics of linear algebra that has several applications across the domain. In this article, we will discuss different types of matrix such as square matrix, row matrix, column matrix, and many more.

In Linear Algebra, Matrices are one of the most important topics of mathematics. The application of matrix is not just limited to mathematical solving problems; it has its applications across all the domains such as Data Science, Machine Learning, Cryptography, Wireless Communication, and many more. In this article, we will discuss different types of matrix with examples. In the final section, we will also discuss some special matrices, such as the Idempotent Matrix, Involutory Matrix, and Orthogonal Matrix.

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Table of Content

• What is Matrix?
• Types of Matrix
  • Singleton Matrix
  • Row Matrix
  • Column Matrix
  • Null/Zero Matrix
  • Square Matrix
  • Diagonal Matrix
  • Scalar Matrix
  • Identity Matrix
  • Triangular Matrix
    • Upper Triangular Matrix
    • Lower Triangular Matrix
  • Symmetric Matrix
  • Skew-Symmetric Matrix

• Some Special Matrix
  • Idempotent Matrix
  • Involutory Matrix
  • Orthogonal Matrix

What is Matrix?

A matrix is a rectangular arrangement of data in rows and columns to form an array.

• The horizontal arrangement is called a row.
• A vertical arrangement is called a column.

Notation: Any matrix having m rows and n columns is given by: [A]_{m x n}.

• Each member of the matrix is called an element of the matrix.
• Order of a Matrix: If a matrix has m rows and n columns, then it is said to be a matrix of order mn.

Now, moving forward, we will know the different types of matrix.

Types of Matrix

Singleton Matrix

A matrix having only one element is known as a singleton matrix.

• A= [a_ij]_mn is a singleton if i = j = 1.
• Example: [1], [0], [9]

Row Matrix

A matrix having only one row is called a row matrix.

• It can have any number of the column.
• A = [a_ij]_mn is a row matrix if i = 1 and 1 <= j <= n.
• Example: A = [1, 2, 3, 4, 5], B = [ -3, -6, -9]

Column Matrix

A matrix having only one column is called a column matrix.

• It can have any number of rows.
• A= [a_ij]_mn is a row matrix if 1 <= i <= m and j = 1.
• Example: 

Null Matrix/Zero Matrix:

If all the entries of a matrix are zero, it is called a null matrix.

• It can be a square or rectangular matrix.
• A = [a_ij]_mn is called a zero matrix if a_ij = 0.
• Example:

Square Matrix

Any matrix is said to be a square matrix if it has the same number of rows and a same number of columns.

• A= [a_ij]_mn is a square matrix if m = n.
• Example:

Note: The vector of values from the diagonal of the matrix from the top left to the bottom right is known as a Principal diagonal. 

Diagonal Matrix

If in a square matrix, all the elements of a principal diagonal are non-zero and all the other entries are zero, then it is called a diagonal matrix.

• A= [a_ij]_mn is a diagonal matrix if a_ij = 0 for all i != j (i not equal to j) and for i = j, a_ij can take any value from real numbers.
• Example

Note: Null matrix is also a diagonal matrix.

Scalar Matrix

It is a particular case of a diagonal matrix in which all the elements of the diagonal are identical.

• In simple terms, if all the elements of a diagonal matrix are equal, it is called a scalar matrix.
• A= [a_ij]_mn is a scalar matrix if a_ij = 0 for all i != j (i not equal to j) and a_ij = a for all i = j, where a is a constant

Identity Matrix

It is a particular case of the scalar matrix. If all the non-zero elements of the scalar matrix are equal to 1, then it is termed as an Identity matrix.

• A= [a_ij]_mn is a scalar matrix if a_ij = 0 for all i != j (i not equal to j) and a_ij = 1 for all i = j
• Example

Triangular Matrix

A square matrix that has all the values in the upper-right or lower left and the remaining values are filled with zero, then such matrices are called Triangular matrix.

These are of two types:

• Upper Triangular Matrix
• Lower Triangular Matrix

Note: Zero/Null Matric is also a type of Triangular Matrix.

Upper Triangular Matrix

A square matrix in which all the elements below the principal diagonal are zero is known as a upper triangular matrix.

• A= [a_ij]_mn is a scalar matrix if a_ij = 0, when i > j.
• Example:

Lower Triangular Matrix

A square matrix in which all the elements above the principal diagonal are zero is known as Lower Triangular Matrix.

• A= [a_ij]_mn is a scalar matrix if a_ij = 0, when i < j.
• Example:

Symmetric Matrix

A square matrix that is equal to its transpose is called a symmetric matrix, i.e., if A is any square matrix, then A is said to be a symmetric matrix if and only if:

A = A^T

Representation:

If A= [a_ij]_mn   is a symmetric matrix, then a_ij = a_ji, where:

• 1<= i <= n, 1 <= j <= n and n belong Natural Number.
• a_ij is an element at the position (i, j), which is i^th row and j^th column.

Example:

Also Read: All About Symmetric Matrix

Skew-Symmetric Matrix

A square matrix that is equal to its transpose is called a symmetric matrix i.e., if A is any square matrix, then A is said to be a symmetric matrix if and only if:

A = -A^T 

Representation

If A= [a_ij]_mn   is a symmetric matrix, then a_ij = – a_ji, where:

• 1<= i <= n, 1 <= j <= n and n belong Natural Number.
• a_ij is an element at the position (i, j), which is i^th row and j^th column.

Example:

Special Matrices

Idempotent Matrix

A square matrix A is said to be Idempotent if:

A² = A

Example:

Involutory Matrix

A square matrix A is said to be Involutory if:

A² = I

where I is an identity matrix.

Example:

Orthogonal Matrix

A square matrix A is said to be an orthogonal matrix if it satisfies the following:

A.A^T = I

where  A^T  is the transpose of a matrix.

and I is an Identity matrix. 

Example: 

Conclusion

In this article, we have discussed different types of matrices, such as Singleton Matrix, Row Matrix, Column Matrix, Null Matrix, Square Matrix, Diagonal Matrix, Triangular Matrix, Symmetric Matrix, and Skew-Symmetric Matrix.
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