Nilpotent matrix – properties and example


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.

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 ≠ 0So [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.

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.

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