Nilpotent Matrix

A square matrix M with dimension n × n is called nilpotent if there exists a positive integer k ≤ n such that:

M^k = O

where,

• O represents the zero matrix of the same dimensions as M.
• The smallest integer k that satisfies the equation is called the index or degree of the nilpotent 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.

Nilpotent Matrix Examples

• The matrix given below is a nilpotent matrix of order "2 × 2."

• The 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

The following are some important properties of a nilpotent matrix:

• A nilpotent matrix is always a square matrix of order "n × n."

• The 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.

• The null matrix is the only diagonalizable nilpotent matrix.

• A nilpotent matrix is a scalar matrix.

• Any triangular matrix with zeros on the principal diagonal is also nilpotent.

• Eigenvalues of a nilpotent matrix are always equal to zero.

Related Articles

• Determinant of a Matrix
• Inverse of a Matrix
• Matrix Addition and Scalar Multiplication

Examples of 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.