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.

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.

Nilpotent Matrix Examples

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

[Tex]A = \left[\begin{array}{cc} 2 & -4\\ 1 & -2 \end{array}\right] [/Tex]

[Tex]A^{2} = A × A = \left[\begin{array}{cc} 2 & -4\\ 1 & -2 \end{array}\right] \times \left[\begin{array}{cc} 2 & -4\\ 1 & -2 \end{array}\right] [/Tex]

[Tex]A^{2} =\left[\begin{array}{cc} (4-4) & (-8+8)\\ (2-2) & (-4+4) \end{array}\right] [/Tex]

[Tex]A^{2} = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right] = O [/Tex]

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

[Tex]B = \left[\begin{array}{ccc} 0 & -4 & 2\\ 0 & 0 & 4\\ 0 & 0 & 0 \end{array}\right] [/Tex]

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.

[Tex]B^{2} =\left[\begin{array}{ccc} 0 & -4 & 2\\ 0 & 0 & 4\\ 0 & 0 & 0 \end{array}\right]\times\left[\begin{array}{ccc} 0 & -4 & 2\\ 0 & 0 & 4\\ 0 & 0 & 0 \end{array}\right] [/Tex]

[Tex]B^{2} =\left[\begin{array}{ccc} 0 & 0 & -16\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] [/Tex]

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

[Tex]B^{3} =\left[\begin{array}{ccc} 0 & 0 & -16\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]\times\left[\begin{array}{ccc} 0 & 0 & -16\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] = O [/Tex]

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

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.

Articles Related to Nilpotent Matrix:

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

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

[Tex]P = \left[\begin{array}{ccc} 1 & -3 & 1\\ 2 & 0 & 2\\ -1 & 3 & -1 \end{array}\right] [/Tex]

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.

[Tex]P^{2} = \left[\begin{array}{ccc} 1 & -3 & 1\\ 2 & 0 & 2\\ -1 & 3 & -1 \end{array}\right] \times\left[\begin{array}{ccc} 1 & -3 & 1\\ 2 & 0 & 2\\ -1 & 3 & -1 \end{array}\right] [/Tex]

[Tex]P^{2} = \left[\begin{array}{ccc} (1-6-1) & (-3+0+3) & (1-6-1)\\ (2+0-2) & (-6+0+6) & (2+0-2)\\ (-1+6+1) & (3+0-3) & (-1+6+1) \end{array}\right] [/Tex]

[Tex]P^{2} = \left[\begin{array}{ccc} -6 & 0 & -6\\ 0 & 0 & 0\\ 6 & 0 & 6 \end{array}\right] [/Tex]

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

[Tex]P^{3} = \left[\begin{array}{ccc} 1 & -3 & 1\\ 2 & 0 & 2\\ -1 & 3 & -1 \end{array}\right]\times\left[\begin{array}{ccc} -6 & 0 & -6\\ 0 & 0 & 0\\ 6 & 0 & 6 \end{array}\right] [/Tex]

[Tex]P^{3}= \left[\begin{array}{ccc} (-6-0+6) & 0 & (-6-0+6)\\ (-12+0+12) & 0 & (-12+0+12)\\ (6+0-6) & 0 & (6+0-6) \end{array}\right] [/Tex]

[Tex]P^{3}=\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] = O [/Tex]

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.

[Tex]M = \left[\begin{array}{cc} 5 & -5\\ 5 & -5 \end{array}\right] [/Tex]

Solution:

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

[Tex]M^{2} = \left[\begin{array}{cc} 5 & -5\\ 5 & -5 \end{array}\right]\times\left[\begin{array}{cc} 5 & -5\\ 5 & -5 \end{array}\right] [/Tex]

[Tex]M^{2} = \left[\begin{array}{cc} (25-25) & (-25+25)\\ (25-25) & (-25+25) \end{array}\right] [/Tex]

[Tex]M^{2} = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right] = O [/Tex]

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.

[Tex]A = \left[\begin{array}{ccc} 0 & 15 & -17\\ 0 & 0 & 12\\ 0 & 0 & 0 \end{array}\right] [/Tex]

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.

[Tex]A^{2} = \left[\begin{array}{ccc} 0 & 15 & -17\\ 0 & 0 & 12\\ 0 & 0 & 0 \end{array}\right] \times \left[\begin{array}{ccc} 0 & 15 & -17\\ 0 & 0 & 12\\ 0 & 0 & 0 \end{array}\right] [/Tex]

[Tex]A^{2} = \left[\begin{array}{ccc} 0 & 0 & 180\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] [/Tex]

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

[Tex]A^{3} = A^{2} × A [/Tex]

[Tex]A^{3} = \left[\begin{array}{ccc} 0 & 0 & 180\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]\times\left[\begin{array}{ccc} 0 & 15 & -17\\ 0 & 0 & 12\\ 0 & 0 & 0 \end{array}\right] [/Tex]

[Tex]A^{3} = \left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] = O [/Tex]

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

What is meant by a nilpotent matrix?

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

Is a nilpotent matrix singular?

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

What is the order of a Nilpotent Matrix?

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.

How to find whether a matrix is nilpotent or not?

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.