:: Practise problems for triangular-block nilpotent form

Use Sage to do your computations for you. See Appendix B.

1. Suppose \(A\) is a \(13 \times 13\) nilpotent matrix. Let \(N\) represent the triangular-block nilpotent form of \(A\) with elementary nilpotent blocks arranged by descending size, as usual. Use the following information to determine the exact form of \(N\). \[\begin{gather*} \operatorname{nullity} A = 5 \qquad A^4 = \mathbf{0} \qquad A^3 \neq \mathbf{0} \\ \operatorname{rank} (A^3) = 2 \qquad \operatorname{rank} (A^2) = 4 \qquad \operatorname{rank} A = 8 \end{gather*} \]
2. For each of the nilpotent matrices from the practise problems for elementary nilpotent form that were not similar to an elementary nilpotent form matrix, determine a transition matrix \(P\) so that \(P^{-1} A P\) is in triangular-block nilpotent form, and also determine the form matrix \(P^{-1} A P\) without computing \(P^{-1}\). (In fact, you shouldn't have to do any computations at all to determine the form matrix.)
   • To determine \(P\), use either Procedure 33.4.2 (see also Example 33.5.3) or Procedure 33.4.3 (see also Example 33.5.4).
   • To determine \(P^{-1} A P\), see Remark 33.5.5.

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