Nilpotent Matrix

Nilpotent Matrices are special types of square matrices, they are special because the product of a Nilpotent Matrix with itself is equal to a null matrix. Let’s take a  square matrix A of order n × n it is considered a nilpotent matrix if A^k = 0. Here k is always less than equal to n. 

In this article, we will learn about the Nilpotent Matrix in detail.

Nilpotent Matrix Definition

A square matrix is said to be a nilpotent matrix if the product of the matrix with itself is equal to a null matrix. In simple words, a square matrix “A” of order “n × n” is said to be nilpotent if “A^k = O,” where “O” is a null matrix of order “n × n” and “k” is a positive integer less than or equal to n. A nilpotent matrix is a square matrix that has an equal number of rows and columns and also it satisfies matrix multiplication. For example, if “P” is a nilpotent matrix of order “2 × 2,” then its square must be a null matrix. If “P” is a nilpotent matrix of order “3 × 3,” then either its square or cube must be a null matrix.

Examples of a Nilpotent Matrix

• Matrix given below is a nilpotent matrix of order “2 × 2.”

• Matrix given below is a nilpotent matrix of order “3 × 3.”

As the order of the given matrix is “3 × 3,” then either its square or cube of the matrix must be a null matrix if it is nilpotent. Now, let us find its square first.

Square of the matrix is not a null matrix. So, let us find its cube now.

We can see that cube of the matrix “B” is a null matrix. So, the given matrix “B” is nilpotent.

Properties of a Nilpotent Matrix

Following are some important properties of a nilpotent matrix:

• A nilpotent matrix is always a square matrix of order “n × n.”
• Nilpotency index of a nilpotent matrix of order “n × n” is always equal to either n or less than n.
• Both the trace and the determinant of a nilpotent matrix are always equal to zero.
• As the determinant of a nilpotent matrix is zero, it is not invertible.
• Null matrix is the only diagonalizable nilpotent matrix.
• A nilpotent matrix is a scalar matrix.
• Any triangular matrices with zeros on the principal diagonal are also nilpotent.
• Eigenvalues of a nilpotent matrix are always equal to zero.

Solved Examples on Nilpotent Matrix

Example 1: Verify whether the matrix given below is nilpotent or not.

Solution:

Order of the given matrix is “3 × 3.” If the given matrix is nilpotent, then either its square or cube of the matrix must be a null matrix.

Now, let us find its square first.

Square of the matrix is not a null matrix. So, let us find its cube now.

We can see that cube of the matrix “P” is a null matrix. So, the given matrix “P” is nilpotent.

Example 2: Verify whether the matrix given below is nilpotent or not.

Solution:

The order of the given matrix is “2 × 2.” If the given matrix is nilpotent, then its square must be a null matrix.

We can see that square of the matrix “M” is a null matrix. So, the given matrix “M” is nilpotent.

Example 3: Determine whether the matrix given below is nilpotent or not.

Solution:

Order of the given matrix is “3 × 3.” If the given matrix is nilpotent, then either its square or cube of the matrix must be a null matrix. Now, let us find its square first.

The square of the matrix is not a null matrix. So, let us find its cube now.

We can see that cube of the matrix “A” is a null matrix. So, the given matrix “A” is nilpotent.

FAQs on Nilpotent Matrix

Question 1: What is meant by a nilpotent matrix?

Answer:

A square matrix is said to be a nilpotent matrix if the product of the matrix with itself is equal to a null matrix.

Question 2: Is a nilpotent matrix singular?

Answer:

Yes, a nilpotent matrix is singular, as the determinant of the nilpotent matrix is always zero.

Question 3: What is the order of a Nilpotent Matrix?

Answer:

The order of a nilpotent matrix is determined by the number of rows and columns that it has. In general, a nilpotent matrix has an equal number of rows and columns because it is a square matrix.

Question 4: How to find whether a matrix is nilpotent or not?

Answer:

To find whether the given matrix is nilpotent or not, we have to check if the product of the matrix with itself is a null matrix. For example, if “P” is a nilpotent matrix of order “2 × 2,” then its square must be a null matrix. If “P” is a nilpotent matrix of order “3 × 3,” then either its square or cube must be a null matrix.