Nilpotent Matrix - Examples

Examples

The matrix

M = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}

is nilpotent, since M2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix

N = \begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix}

is nilpotent, with

N^2 = \begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^3 = \begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^4 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}.

Though the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, the matrices

\begin{bmatrix} 6 & -9 \\ 4 & -6 \end{bmatrix}\qquad\text{and}\qquad \begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix}

both square to zero, though neither matrix has zero entries.

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