Nilpotent Matrices

  • Thread starter the_kid
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  • #1

Homework Statement

Suppose that N is a nilpotent mxm matrix, N[itex]^{m}[/itex]=0, but N[itex]^{m'}[/itex][itex]\neq[/itex]0 for m'<m. Show that there exists a basis in which it takes the form of a single Jordan block with vanishing diagonal elements. Prove that your basis set is linearly independent.

Homework Equations

The Attempt at a Solution

So I've recognized the fact that N[itex]^{m'}[/itex][itex]\neq[/itex]0 for m'<m means that N[itex]^{m'}[/itex] does not annihilate every vector in V. I'm just not really sure where go from here...

Answers and Replies

  • #2
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Pick a basis for the vectors that are annihilated by N. Then add a basis for the vectors that are annihilated by N^2 but not by N. Continue. Eventually you'll get a basis for the whole space, right? What does N look like in that basis?