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."

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

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]

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

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

• The matrix given below is a nilpotent matrix of order "3 × 3."

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

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.

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] = =\left[\begin{array}{ccc} 0 & 0 & -16\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]

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

B^{3} =\left[\begin{array}{ccc} 0 & 0 & -16\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]\times\left[\begin{array}{ccc} 0 & -4 & 2\\ 0 & 0 & 4\\ 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right] = O

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.

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

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.

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]

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]

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

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

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]

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]

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

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.

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

Solution:

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

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]

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

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

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.

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

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.

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]

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

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

A^{3} = A^{2} × A

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]

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

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