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
1. \[A=\begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}\]
Matrix A is a nilpotent matrix of index 2. It means that A ≠ O and A2 = O.
2. \[B=\begin{bmatrix} 2 &-1 \\ 4& -2 \end{bmatrix}\]
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
1. \[C=\begin{bmatrix} 5 &-3 &2 \\ 15& -9 &6 \\ 10&-6 & 4 \end{bmatrix}\]
Matrix C is a nilpotent matrix of index 2. It means that C ≠ O and C2 = O.
2. \[D=\begin{bmatrix} 1 &1 &3 \\ 5& 2 &6 \\ -2&-1 & -3 \end{bmatrix}\]
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.
Correction for Matrix D: D!=0,D^2!=0, D^3=0 instead of D!=0,D^2=0,D^3!=0
Great explanation of nilpotent matrices! I really appreciated the examples you provided—they made the concept much clearer. The properties you outlined were also helpful for understanding their significance in linear algebra. Looking forward to more posts like this!