Nilpotent Matrix – Definition and Example

Because the product of a Nilpotent Matrix and itself equals a null matrix, Nilpotent Matrix is a unique variety of square matrix. Let A be a square matrix of order n × n. If Ak = 0, A is a nilpotent matrix. In this case, k is never equal to n.

We shall study the Nilpotent Matrix in detail in this post.

The Definition of Nilpotent Matrix

Nilpotent matrix is a square matrix which will produce a null matrix if we multiply it with itself.

A square matrix is a nilpotent matrix if Ak = 0, where A is a square matrix with a dimension (order) of n x n, and k is the exponent. The value of k is less than or equal to the dimension or order of n.


A nilpotent matrix is a square matrix with n x n that fulfills Ak = 0, where kn.

It is a bit different with an idempotent matrix.

Example of Nilpotent Matrix

In order to make us understand better, let us put it into practice.

Assume matrix A is a square matrix with the order of 2

nilpotent matrix

Properties of Nilpotent Matrix

  • An order n × n square matrix is the nilpotent matrix.
  • A nilpotent matrix with an order of n × n has an index that is either n or less than n.
  • A nilpotent matrix has eigenvalues that are all equal to zero.
  • A nilpotent matrix’s trace, or determinant, is always zero.
  • A scalar matrix is the nilpotent matrix.
  • There is no way to invert the nilpotent matrix.

Nilpotent Matrix Index Calculation

The smallest value of “k” is known as the index of the Nilpotent matrix. The definition states that if a square matrix [A] is a Nilpotent matrix, it will satisfy the equation Ak = 0 for certain positive values of “k.”

Therefore, all you have to do is multiply matrix [A] by the same matrix again until you obtain a zero matrix, or null matrix (0), in order to find the index of the nilpotent matrix. Assume, for instance, that after multiplying matrix [A] by k times, you obtained Ak = 0.

As a result, that integer value k will be the index of that Nilpotent matrix [A].

The n x n Nilpotent matrix’s index will always be at most n times its value. Therefore, you will need to multiply the maximal matrix by n (the matrix’s order).

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