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!
This post was really helpful in breaking down the concept of nilpotent matrices! The examples you provided made it much easier to grasp the properties. Looking forward to more math insights on your blog!
This post on nilpotent matrices is really interesting! I appreciated the clear explanations and the examples you provided. It helped me understand their properties better, especially in the context of linear transformations. Looking forward to reading more about similar topics!
Great explanation of nilpotent matrices! I particularly appreciated the clear examples you provided, which made the concept much easier to grasp. The properties section was insightful as well—it’s fascinating how such matrices can simplify complex problems in linear algebra. Looking forward to more posts like this!
Great post! I found the explanation of nilpotent matrices really clear and the examples helpful for understanding the concept better. I appreciate how you linked the properties to practical applications in electrical engineering. Looking forward to more posts like this!