Nilpotent Matrix

Nilpotent Matrix is a square matrix such that the product of the matrix with itself is equal to a null matrix. A square matrix M of order n × n is termed as a nilpotent matrix if M^k = 0. Here k is the exponent of the nilpotent matrix and is lesser than or equal to the order of the matrix( k < n). The order of a nilpotent matrix is n × n, and it easily satisfies the condition of matrix multiplication.

Let us learn more bout the nilpotent matrix, properties of the nilpotent matrix, and also check the examples, FAQs.

A nilpotent matrix is a square matrix A such that A^k = 0. Here k is called the index or exponent of the matrix, and 0 is a null matrix with the same order as that of matrix A. The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not. Further, the exponent of a nilpotent matrix is lesser than or equal to the order of the matrix (k < n).

The nilpotent matrix is a square matrix with an equal number of rows and columns and it satisfies the condition of matrix multiplication. For a square matrix of order 2 x 2, to be a nilpotent matrix, the square of the matrix should be a null matrix, and for a square matrix of 3 x 3, to be a nilpotent matrix, the square or the cube of the matrix should be a null matrix.

Let us check a few examples, for a better understanding of the working of a nilpotent matrix.

Examples of Nilpotent Matrix

1. An example of 2 × 2 Nilpotent Matrix is A = $\begin{bmatrix}4&-4\\4&-4\end{bmatrix}$

A² = $\begin{bmatrix}4&-4\\4&-4\end{bmatrix}$ × $\begin{bmatrix}4&-4\\4&-4\end{bmatrix}$

= $\begin{bmatrix}4×4+(-4)×4&4×(-4)+(-4)×(-4)\\4×4 + (-4)× 4&4×(-4) + (-4)×(-4)\end{bmatrix}$

= $\begin{bmatrix}16 - 16&-16 + 16\\16 - 16&-16 + 16\end{bmatrix}$

= $\begin{bmatrix}0&0\\0&0\end{bmatrix}$

2. A n-dimensional triangular matrix with zeros along the main diagonal can be taken as a nilpotent matrix.

A = $\begin{bmatrix}0&3&2&1\\0&0&2&2\\0&0&0&3\\0&0&0&0\end{bmatrix}$

A² = $\begin{bmatrix}0&0&6&12\\0&0&0&6\\0&0&0&0\\0&0&0&0\end{bmatrix}$

A³ = $\begin{bmatrix}0&0&0&18\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$

A⁴ = $\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$

3. Also, a matrix without any zeros can also be referred as a nilpotent matrix. The following is a general form of a non-zero matrix, which is a nilpotent matrix.

A = $\begin{bmatrix}p&p&p&p\\q&q&q&q\\r&r&r&r\\-(p + q + r)&-(p + q + r)&-(p + q + r)&-(p + q + r)\end{bmatrix}$

Let A = $\begin{bmatrix}3&3&3\\4&4&4\\-7&-7&-7\end{bmatrix}$

A² = $\begin{bmatrix}3&3&3\\4&4&4\\-7&-7&-7\end{bmatrix}$ × $\begin{bmatrix}3&3&3\\4&4&4\\-7&-7&-7\end{bmatrix}$

= $\begin{bmatrix}3×3+3×4+3×(-7)&3×3+3×4+3×(-7)&3×3+3×4+3×(-7)\\4×3+4×4+4×(-7)&4×3+4×4+4×(-7)&4×3+4×4+4×(-7)\\(-7)×3+(-7)×4+(-7)×(-7)&(-7)×3+(-7)×4+(-7)×(-7)&(-7)×3+(-7)×4+(-7)×(-7)\end{bmatrix}$

= $\begin{bmatrix}9+12-21&9+12-21&9+12-21\\12 + 16 - 28&12 + 16 - 28&12 + 16 - 28\\-21 -28 + 49&-21 -28 + 49&-21 -28 + 49\end{bmatrix}$

= $\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}$

= $\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}$

Nilpotent matrix is a square matrix, which on multiplying with itself results in a null matrix. The following are some of the important properties of nilpotent matrices.

• The nilpotent matrix is a square matrix of order n × n.
• The index of a nilpotent matrix having an order of n ×n is either n or a value lesser than n.
• All the eigenvalues of a nilpotent matrix are equal to zero.
• The determinant or the trace of a nilpotent matrix is always zero.
• The nilpotent matrix is a scalar matrix.
• The nilpotent matrix is non-invertible.

Related Topics

The following topics help in a better understanding of the nilpotent matrix.

• Singular Matrix
• Unitary Matrix
• Hermitian Matrix
• Orthogonal Matrix
• Scalar Matrix