# Nilpotent Matrix – Definition and Example

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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.

Thus,

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

## 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).