Use Sage to do your computations for you.
See Appendix B.5: Triangular block form.
Carry out steps a, b, c below for each of the following matrices.
- \(A = \begin{bmatrix} -5 & 3 & 1 \\ -4 & 2 & 1 \\ -4 & 3 & 0 \end{bmatrix}\)
- \(A = \begin{bmatrix} -3 & 6 & 3 & 2 \\ -2 & 3 & 2 & 2 \\ -1 & 3 & 0 & 1 \\ -1 & 1 & 2 & 0 \end{bmatrix}\)
NOTE:
These are the same matrices as in the second problem from the practise problems for scalar-triangular form.
If you have not already done so, you should complete that problem first since the tasks below mostly just ask you to interpret the calculation results obtained
in that problem.
- Verify that the characteristic polynomial \(c_A(\lambda)\) factors completely as in the hypothesis of Theorem 30.5.1: Triangular block form of a matrix.
- Explicitly verify that the conclusions of Statement 2 of Theorem 30.5.5: More properties of generalized eigenspaces hold for
\(A\).
- Determine a transition matrix \(P\) such that \(P^{-1} A P\) is in triangular block form. Compute that triangular block form matrix \(P^{-1} A P\) without explicitly computing \(P^{-1}\).
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