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Trivial question!

  1. Nov 5, 2006 #1
    Can a zero matrix be considered as nilpotent matrix?
    Zero matrix raised to any positive power is zero matrix, so can it be considered nilpotent (with index of nilpotency being 1)? I have read the definition of the nilpotent matrix and it doesn't that say that a matrix has to different from 0, but I'm still confused...
  2. jcsd
  3. Nov 5, 2006 #2
    sure, you can say that the zero matrix is nilpotent, but that would be considered the trivial case. i.e., if you are asked to find a nilpotent matrix satisfying some properties, using the zero matrix will probably not get you credit for solving the question.
  4. Nov 6, 2006 #3


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    Staff Emeritus
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    Why are you confused? Another characterization of "nilpotent" is that a matrix is nilpotent if and only if its eigenvalues are all 0. That is certainly true of the 0 matrix. The 0 matrix is definitely nilpotent.
  5. Nov 8, 2006 #4
    Thank you!
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