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Recipe

Confirm an operator is nilpotent and find the smallest k

Use when: you're given a matrix N and need to determine whether some power N^k equals the zero matrix, and if so, the smallest such k

Goal: either the smallest k with N^k=\mathbf 0, confirming nilpotency, or a specific power that stays nonzero, ruling nilpotency out

Taught in Nilpotent operators, first look

Quick recipe

  1. Compute N^2 and check it against the zero matrix
  2. If N^2 isn't zero, keep multiplying by N one power at a time
  3. Stop once a power reaches zero, or once N^n is checked and still isn't
  4. Report the smallest k, not just any zero power
  1. 1

    Multiply N by itself once and compare every entry of the result to zero.

    Example

    For N=\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}: N^2=\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix} which is not the zero matrix.

    Why: N^2 is the cheapest power to check first, and it already settles nilpotency for plenty of small examples.

  2. 2

    Compute N^3=N^2\cdot N, check it against zero, and continue to N^4=N^3\cdot N and beyond if needed, stopping as soon as a power is zero.

    Example

    Continuing the example above: N^3=N^2\cdot N=\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}=\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}

    Why: reusing the previous power avoids recomputing N multiplied by itself from scratch at every step.

  3. 3

    For an n\times n matrix, a genuinely nilpotent N always reaches the zero matrix by N^n at the latest, so checking up through N^n settles the question either way.

    • some power up to N^n equals the zero matrix — N is nilpotent, and the smallest such power is the answer.
    • N^n itself is still nonzero — N is not nilpotent, and no larger power needs to be checked.

    ⚡ Shortcut

    If N is triangular with every diagonal entry already zero, it's automatically nilpotent, with no powers to compute at all. For N=\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}, the zero diagonal alone guarantees N^3=\mathbf 0, matching a direct computation exactly.

    Why: a genuinely nilpotent operator on an n-dimensional space is guaranteed to collapse to zero by the n-th power at the latest.

  4. 4

    Once a zero power is found, confirm the power just before it was still nonzero, so the reported k is genuinely the smallest one.

    Example

    Since N^2\ne\mathbf 0 but N^3=\mathbf 0 in the running example, the smallest k is 3, not some larger power that also happens to work.

    Why: any power past the smallest one that reaches zero is also zero, so reporting a larger k would understate how quickly N collapses.

Common mistakes

  • Stopping at the first power checked, without recognizing that a later power might still be needed.
  • Reporting a power that also equals zero, but skipping over a smaller one that already worked.
  • Assuming a nonzero N^2 already rules out nilpotency, without checking any higher power.
  • Forgetting that a triangular matrix with a zero diagonal is automatically nilpotent, and recomputing powers by hand anyway.