Nilpotent matrix - properties and example

SHARE:

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

nilpotent matrix

What is Nilpotent matrix?

Nilpotent matrix: Any square matrix [A] is said to be Nilpotent matrix if it satisfy the condition [Ak] = 0 and [Ak-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 Ak = 0 for all k ≥ n. For example a 2 x 2 square matrix [A] will be Nilpotent of degree 2 if A2 = 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

example of 4 x 4 nilpotent matrix

Here in this triangular matrix all its diagonal elements are zero. Also here A4 = 0 but A3 ≠ 0. So [A] will be nilpotent matrix of order or degree 4.

example of 3 x 3 nilpotent matrix
Here in this 3 x 3 matrix B2 = 0 but B1 ≠ 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, Ak = 0 ).
  • The characteristics polynomial of [A] will be det(xI - A) = xn
  • The minimal polynomial of [A] will be xk provide k ≤ n.
  • The only (complex) Eigen value of [A] is zero (0).
  • Trace (Ak) = 0 for all k > 0 i.e, sum of all diagonal entries of [Ak] will be zero.
  • 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 Ak = 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 Ak = 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.

Read More:

COMMENTS

BLOGGER: 1
  1. Thanks for such an informative article...
    Nicely explained.. everything is Cristal clear...

    ReplyDelete

Name

Electrical Basics,8,Electrical Machine,3,Electrical Q & A,2,Electronics,3,Instrumentation,2,Mathematics,3,Power Factor,3,Power System,8,
ltr
item
Electrical-Technology | All about Electrical Engineering: Nilpotent matrix - properties and example
Nilpotent matrix - properties and example
In general any triangular matrix with zeros along with its main diagonal is Nilpotent matrix.
https://blogger.googleusercontent.com/img/a/AVvXsEjelfqbeIKSviust2q7TukKRBZPHAJDAvzExalClsV78wGS7Fwwn0yudoAqQY_GIHBJMz5jUXoekfjM1PahoB2ZhLm9OehPvrtojvol24h_3YiCk6rjkinatUhIXLuGRmyDh_QJFJgxdFuYxLOa214G_8WevsVBem1SEMfRAq6-MUCnUBFB5WYneYA_=w400-h151
https://blogger.googleusercontent.com/img/a/AVvXsEjelfqbeIKSviust2q7TukKRBZPHAJDAvzExalClsV78wGS7Fwwn0yudoAqQY_GIHBJMz5jUXoekfjM1PahoB2ZhLm9OehPvrtojvol24h_3YiCk6rjkinatUhIXLuGRmyDh_QJFJgxdFuYxLOa214G_8WevsVBem1SEMfRAq6-MUCnUBFB5WYneYA_=s72-w400-c-h151
Electrical-Technology | All about Electrical Engineering
https://www.electrical-technology.com/2021/10/nilpotent-matrix.html
https://www.electrical-technology.com/
https://www.electrical-technology.com/
https://www.electrical-technology.com/2021/10/nilpotent-matrix.html
true
1120548955437985836
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED STEP 1: Share to a social network STEP 2: Click the link on your social network Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy Table of Content