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).
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Nilpotent matrix Examples
The examples of 2 x 2 nilpotent matrices are
1.
![\[A=\begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-5892e998f4284cba613db119bffb8152_l3.png "Rendered by QuickLaTeX.com")
Matrix A is a nilpotent matrix of index 2. It means that A ≠ O and A2 = O.
2.
![\[B=\begin{bmatrix} 2 &-1 \\ 4& -2 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-bd30234b7e1db3366181ac853f89db62_l3.png "Rendered by QuickLaTeX.com")
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.
![\[C=\begin{bmatrix} 5 &-3 &2 \\ 15& -9 &6 \\ 10&-6 & 4 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-2394beb66c050494b3c214a4fc053f16_l3.png "Rendered by QuickLaTeX.com")
Matrix C is a nilpotent matrix of index 2. It means that C ≠ O and C2 = O.
2.
![\[D=\begin{bmatrix} 1 &1 &3 \\ 5& 2 &6 \\ -2&-1 & -3 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-e4ba52f94341d38e14954f514f7346ee_l3.png "Rendered by QuickLaTeX.com")
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