Check if matrix is nilpotent (A^k = 0)
A nilpotent matrix becomes the zero matrix when raised to some positive power. The index of nilpotency is the smallest such power. Nilpotent matrices have only zero as their eigenvalue.
A nilpotent matrix is a square matrix A for which some power A^k equals the zero matrix for some positive integer k. The smallest such k is called the index of nilpotency. Strictly upper (or lower) triangular matrices with zeros on the diagonal are nilpotent.
All eigenvalues are 0. Characteristic polynomial = λⁿ.
Since all eigenvalues are 0, trace is always 0.
For n×n matrix, nilpotency index is at most n.
Example: N = [[0,1],[0,0]] (strictly upper triangular)
1. N² = [[0,0],[0,0]] = 0 ✓
2. N is nilpotent with index 2 ✓
3. N projects vectors onto y-axis, then kills them.
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