Nilpotent Matrix

A nilpotent matrix is essentially a square matrix in which the product of the matrix and itself is a null matrix. If Mk = 0, a square matrix M of rank n n is called a nilpotent matrix. Here, k is its exponent, which is less than or equivalent to the matrix’s order (k < n). In this article, we will discuss the definition of the nilpotent matrix, its formula, properties and examples. 

Definition

A nilpotent matrix is essentially a square matrix N in linear algebra that 

Nk = 0, where k is a positive integer. It is known as the index of N and is also referred to as the degree of N.

A nilpotent transformation essentially is a linear transformation (L) of a vector space that Lk = 0 for some positive integer k.

Both of these ideas are subsets of the broader concept of nilpotence, which applies to ring elements.

If you raise a square matrix to a reasonably high integer power, you receive the zero matrices as a consequence, which is said to be nilpotent.

Nk = 0

N seems to be the nilpotent matrix, while k is the power exponent that yields the null matrix.

This criterion does not imply that the power of a nilpotent matrix consistently returns to zero, irrespective of the exponent, but rather that the matrix is nilpotent if at least 1 power of the matrix returns a matrix packed with 0s.

The nilpotency index of a nilpotent matrix, on the other hand, is the lowest integer that satisfies the nilpotency requirement. It is also known as a k-index matrix 

Nilpotent matrices have certain properties.

The following properties are shared by all nilpotent matrices:

• A nilpotent matrix’s trace will always be zero.

• In the same way, the determinant of every nilpotent matrix will always be 0. However, the converse is not true, i.e., just because a matrix’s determinant is zero doesn’t mean the matrix is nilpotent.

• The null matrix seems to be the only nilpotent matrix that is diagonalizable. 

• A nilpotent matrix of dimensions n n has a nilpotency index which is always equivalent to or less than n. A 2 2 nilpotent matrix’s nilpotency index is always 2.

• It can’t be inverted.

•  Every triangular matrix containing zeros on the major diagonal is a nilpotent matrix.

• There’s a theorem that claims that if a matrix N is nilpotent, it is invertible (N+I), assuming that I is the Identity matrix. Its inverse matrix may also be obtained using the following formula:

(N +I)-1 = m=0 (-N)m = I-N + N2 – N3 + ….

• Alternatively, if N is a nilpotent matrix, the inverse N-I of the matrix may be constructed using the equation:

(N – I)-1 = m=0 (N )m = I + N + N2 + N3 + ….

• The product of nilpotent matrices may be used to decompose any singular matrix.

• A nilpotent matrix has zero eigenvalues.

= 0   

• Finally, there’s the idea of nilpotent transformation, which describes a linear map L of a vector space in a way that Lk = 0.

Examples

We’ll look at a few instances of nilpotent matrices to get a better understanding of the concept:

A,  2 2 nilpotent matrices are examples.

The nilpotent square matrices of order 2 are as follows:

A = 

Since we gain the zero matrix simply squaring matrix A, then matrix is nilpotent:

Because the null matrix is acquired to the second power, this is a nilpotent matrix with such a nilpotency index of 2.

A 3   3 nilpotent matrix is an illustration of it.

The following three-dimensional square matrix contains nilpotent:

B = 

Even though we don’t get the null matrix when we multiply the matrix by two:

When we calculate the matrix’s cube, we have a matrix with all of the elements equivalent to 0:

As a result, matrix B is just a nilpotent matrix, with a nilpotency index of 3 due to the null matrix being acquired to the third power.

Conclusion

A nilpotent matrix is essentially a square matrix N in linear algebra that 

Nk = 0, where k is a positive integer. It is known as the index of N and is also referred to as the degree of N. In this article, we’ve discussed the topic in detail, its properties and examples. Read the article thoroughly, to grasp the concepts, go through the examples and solve as many questions as possible using the formula.