In general any triangular matrix with zeros along with its main diagonal is Nilpotent matrix.

**Nilpotent matrix:** Any square matrix **[A]** is
said to be Nilpotent matrix if it satisfy 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 square matrix [A] is a Nilpotent matrix of order **n x n**, then there must be **A ^{k} = 0** for all

**k ≥ n**. For example a 2 x 2 square matrix [A] will be Nilpotent of degree 2 if

**A**

^{2}= 2.In general any triangular matrix with zeros along with its main
diagonal is Nilpotent matrix. Nilpotent matrix is also a special case of
convergent matrix.

**[ ##eye## Idempotent matrix and its properties]**

**Example of Nilpotent
matrix**

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

**A**So [A] will be nilpotent matrix of order or degree 4.

^{3}≠ 0.Here in this 3 x 3 matrix

**B**

**but**

^{2}= 0**B**

^{1}

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

**Hence [B] will be nilpotent matrix of order 2.**

**[ ##eye## Power Factor Correction techniques]**

**Properties of Nilpotent
matrix**

Following are some important **properties
of nilpotent matrix**.

- 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 Nilpotent matrix then**[I+A]**and**[I-A]**will be invertible. - All eigen values of Nilpotent matrix will be zero (0).
- If
**[A]**is Nilpotent matrix then 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 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],** following are some
important characteristics observed.

- Square matrix
**[A]**is 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**provide^{k}**k ≤ n**. - The only (complex) Eigen value of
**[A]**is zero (0). **Trace (A**for all^{k}) = 0**k > 0**i.e, sum of all diagonal entries of**[A**will be zero.^{k}]- The only Nilpotent diagonalizable matrix is zero matrix.

**How to find index of
Nilpotent matrix**

According to the definition, if a square matrix **[A]** is 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

**index of Nilpotent matrix**.

So to find the index of Nilpotent matrix, simply keep multiplying
matrix **[A]** with 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 ^{k}
= 0**. Hence the index of that Nilpotent matrix

**[A]**will be that integer value

**k**.

There is guarantee that 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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Thanks for such an informative article...

ReplyDeleteNicely explained.. everything is Cristal clear...