:: Answers to practise problems for triangular-block nilpotent form

  1. The matrix \(N\) has block form \[ N = \begin{bmatrix} N_4 \\ & N_4 \\ & & N_2 \\ & & & N_2 \\ & & & & N_1 \end{bmatrix} \] where \(N_4\) is a \(4 \times 4\) elementary nilpotent block, \(N_2\) is a \(2 \times 2\) elementary nilpotent block, and \(N_1\) is a \(1 \times 1\) elementary nilpotent block: \[\begin{align*} N_4 & = \begin{bmatrix} 0 \\ 1 & 0 \\ & 1 & 0 \\ & & 1 & 0 \end{bmatrix} \text{,} & N_2 & = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \text{,} & N_1 & = \begin{bmatrix} 0 \end{bmatrix} \text{.} \end{align*}\]
  2. Here, answer a. corresponds to problem 1. from the practise problems for elementary nilpotent form, answer b. corresponds to problem 2. from that page, and so on. Note that different possibilities for each transition matrix \(P\) are possible depending on different choices made in the procedures.
    1. Matrix is \(3 \times 3\), degree of nilpotency is \(k = 2\).
      • Using Procedure 33.4.2:      \(P = \begin{bmatrix} 1 & 18 & 1 \\ 0 & 18 & 0 \\ 0 & -6 & 2 \end{bmatrix} \)
      • Using Procedure 33.4.3:      \(P = \begin{bmatrix} 1 & 18 & 7 \\ 0 & 18 & 6 \\ 0 & -6 & 0 \end{bmatrix} \)
      In both cases:      \( P^{-1} A P = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ & & 0 \end{bmatrix} \).
    2. Matrix is similar to \(3 \times 3\) elementary nilpotent form. See the answers for the elementary nilpotent form practise problems for a transition matrix.
    3. Matrix is similar to \(3 \times 3\) elementary nilpotent form. See the answers for the elementary nilpotent form practise problems for a transition matrix.
    4. Matrix is \(4 \times 4\), degree of nilpotency is \(k = 3\).
      • Using Procedure 33.4.2:      \(P = \begin{bmatrix} 1 & 4 & -2 & 13 \\ 0 & -11 & 1 & 0 \\ 0 & 4 & 0 & -3 \\ 0 & -1 & 0 & 1 \end{bmatrix} \)
      • Using Procedure 33.4.3:      \(P = \begin{bmatrix} 1 & 4 & -2 & 9 \\ 0 & -11 & 1 & -11 \\ 0 & 4 & 0 & 3 \\ 0 & -1 & 0 & -1 \end{bmatrix} \)
      In both cases:      \( P^{-1} A P = \begin{bmatrix} 0 \\ 1 & 0 \\ & 1 & 0 \\ & & & 0 \end{bmatrix} \).
    5. Matrix is \(4 \times 4\), degree of nilpotency is \(k = 2\). In this case, both procedures arrive at the same transition matrix (depending on choices made along the way): \[\begin{align*} P & = \begin{bmatrix} 1 & -3 & 0 & 15 \\ 0 & 0 & 1 & 26 \\ 0 & 1 & 0 & -11 \\ 0 & 0 & 0 & -4 \end{bmatrix} \text{,} & P^{-1} A P = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ & & 0 & 0 \\ & & 1 & 0 \end{bmatrix} \text{.} \end{align*}\]
    6. Matrix is \(6 \times 6\), degree of nilpotency is \(k = 4\).
      • Using Procedure 33.4.2:      \(P = \begin{bmatrix} 0 & 2 & -2 & 8 & 0 & 41 \\ 1 & 1 & 0 & 4 & -1 & 20 \\ 0 & 1 & 0 & 2 & 0 & 10 \\ 0 & 0 & 1 & 1 & 0 & 5 \\ 0 & 0 & 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}\)
      • Using Procedure 33.4.3: the same \(P\) matrix as above but with the fifth and sixth columns replaced by their negatives.
      In both cases:      \( P^{-1} A P = \begin{bmatrix} 0 \\ 1 & 0 \\ & 1 & 0 \\ & & 1 & 0 \\ & & & & 0 \\ & & & & 1 & 0 \end{bmatrix} \).
    7. Matrix is similar to \(6 \times 6\) elementary nilpotent form. See the answers for the elementary nilpotent form practise problems for a transition matrix.