What is Nilpotent Matrix? Examples & Properties

A nilpotent matrix (P) is a square matrix, if there exists a positive integer ‘m’ such that Pm = O. In other words, matrix P is called nilpotent of index m or class m if Pm = O and Pm-1 ≠ O. Here O is the null matrix (or zero matrix).

Nilpotent matrix Examples

The examples of 2 x 2 nilpotent matrices are


    \[A=\begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}\]

Matrix A is a nilpotent matrix of index 2. It means that A ≠ O and A2 = O.


    \[B=\begin{bmatrix} 2 &-1 \\ 4& -2 \end{bmatrix}\]

Matrix B is a nilpotent matrix of index 2. It means that B ≠ O and B2 = O.

The examples of 3 x 3 nilpotent matrices are


    \[C=\begin{bmatrix} 5 &-3 &2 \\ 15& -9 &6 \\ 10&-6 & 4 \end{bmatrix}\]

Matrix C is a nilpotent matrix of index 2. It means that C ≠ O and C2 = O.


    \[D=\begin{bmatrix} 1 &1 &3 \\ 5& 2 &6 \\ -2&-1 & -3 \end{bmatrix}\]

Matrix D is a nilpotent matrix of index 3. It means that D ≠ O, D2 = O and D3 ≠ O.

Nilpotent matrix Properties

1. It is a square matrix.

2. All of its eigenvalues are zero.

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