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 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.
Example of Nilpotent matrix
Here in this triangular matrix all its diagonal elements are zero. Also here A^{4} = 0 but A^{3} ≠ 0. So [A] will be 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 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 [IA] 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^{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.