**What is Nilpotent Matrix?**

**Nilpotent matrix:** Any square matrix **[A]** is said to be a Nilpotent matrix if it satisfies the condition** [A ^{k}] = 0** and

**[A**

^{k}-1]**≠**

**0**for some positive integer value of

**k**. Then the least value of such positive integer

**k**is called the

**index (or degree) of**

**nilpotency**. If the square matrix

**[A]**is a Nilpotent matrix of order

**n x n**, then there must be

**A**for all

^{k}= 0**k ≥ n**. For example, a 2 x 2 square matrix

**[A]**will be

**Nilpotent of degree 2**if

**A**In general, any triangular matrix with zeros along with its main diagonal is a Nilpotent matrix. A nilpotent

^{2}= 2.**matrix is also a special case of the**

**convergent matrix.**

**Example of Nilpotent Matrix**

Here in this triangular matrix, all its diagonal elements are zero. Also here **A ^{4} = 0** but

**A**So

^{3}≠ 0.**[A]**will be a nilpotent matrix of order or degree 4.

Here in this 3 x 3 matrix **B**** ^{2} = 0** but

**B**

^{1}

**≠ 0,**although it has no zero diagonal elements.

**Hence [B] will be a**

**nilpotent matrix of order 2.**

**Properties of Nilpotent Matrix**

The following are the important **properties of a nilpotent matrix**.

- A nilpotent matrix is a square matrix and also a singular matrix.
- The determinant and trace of Nilpotent matrix will be zero (0).
- If
**[A]**is a Nilpotent matrix then**[I+A]**and**[I-A]**will be invertible. - All eigenvalues of Nilpotent matrix will be zero (0).
- If
**[A]**is a Nilpotent matrix then the determinant of**[I+A] = 1**, where**I**is**n x n**identity matrix. - The degree or index of any
**n x n**Nilpotent matrix will always be less than or equal to ‘**n**’. - For Nilpotent matrices
**[A]**and**[B]**of order**n x n**, if**AB = BA**then**[AB]**and**[A+B]**will also be Nilpotent matrices. - Every singular matrix can be expressed as the product of Nilpotent matrices.

**Characterization of Nilpotent Matrix**

For any **n x n** square matrix **[A],** the following are some important characteristics observed.

- Square matrix
**[A]**is a Nilpotent matrix of degree**k ≤ n (i.e., A**.^{k}= 0 ) - The characteristics polynomial of
**[A]**will be**det(xI – A) = x**^{n} - The minimal polynomial of
**[A]**will be**x**provided^{k}**k ≤ n**. - The only (complex) Eigenvalue of
**[A]**is zero (0). **Trace (A**for all^{k}) = 0**k > 0**i.e., the sum of all diagonal entries of**[A**will be zero.^{k}]- The only Nilpotent diagonalizable matrix is the zero matrix.

**How to find the index of the Nilpotent Matrix**

According to the definition, if a square matrix **[A]** is a Nilpotent matrix then it will satisfy the equation **A ^{k} = 0 **for some positive values of ‘

**k**’, and such smallest value of ‘

**k**’ is known as the

**index of the Nilpotent matrix**. So to find the index of the Nilpotent matrix, simply keep multiplying matrix

**[A]**with the same matrix until you get a zero matrix or null matrix (0). For example suppose you multiplied matrix

**[A], k**times and then you got

**A**. Hence the index of that Nilpotent matrix

^{k }= 0**[A]**will be that integer value

**k**.

There is a guarantee that the index of **n x n** Nilpotent matrix will be at most the value of **n**. So you will have to multiply the matrix maximum **n** (**order of matrix**) times.

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