NOTE:
In the case of a matrix with maximum degree of nilpotency,
other choices of transition matrix \(P\) than the one presented here are possible.
- \(A\) is \(3 \times 3\) but \(A^2 = \mathbf{0} \implies \) nilpotent but not similar to elementary nilpotent form
- \( P = \begin{bmatrix} 1 & 8 & -2 \\ 0 & 25 & -6 \\ 0 & -79 & 19 \end{bmatrix} \)
-
\( P = \begin{bmatrix} 1 & -6 & -2 & 0 \\ 0 & -12 & -4 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 1 & -8 & -2 \end{bmatrix} \)
- \(A\) is \(4 \times 4\) but \(A^3 = \mathbf{0} \implies \) nilpotent but not similar to elementary nilpotent form
- \(A\) is \(4 \times 4\) but \(A^2 = \mathbf{0} \implies \) nilpotent but not similar to elementary nilpotent form
- \(A\) is \(6 \times 6\) but \(A^4 = \mathbf{0} \implies \) nilpotent but not similar to elementary nilpotent form
-
\( P = \begin{bmatrix}
0 & -3 & 17 & -10 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 13 & -10 & 4 & 2 & 0 \\
0 & 2 & 1 & 0 & 0 & 0 \\
0 & -4 & 3 & 1 & 0 & 0 \\
0 & 6 & -5 & 2 & 1 & 0
\end{bmatrix} \)