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 [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 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

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.

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, A^k = 0 ).

• The characteristics polynomial of [A] will be det(xI – A) = xⁿ

• The minimal polynomial of [A] will be x^k provide k ≤ n.

• The only (complex) Eigen value of [A] is zero (0).

• Trace (A^k) = 0 for all k > 0 i.e, sum of all diagonal entries of [A^k] 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 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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