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Note that other answers are possible besides the ones presented here.
In choosing our basis vectors to create transition matrices,
we often chose parameter values other than 1 in order to clear fractions.
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\(P = \begin{bmatrix} 1 & -3 & 1 \\ -2 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 0 & 2 \\ 0 & 5 & -8 \\ 0 & 0 & 5 \end{bmatrix} \)
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\(P = \begin{bmatrix} -1 & -1 & 1 \\ 2 & 0 & 0 \\ 2 & 4 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 2 & -4 \\ 0 & 5 & -2 \\ 0 & 0 & 5 \end{bmatrix} \)
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\(P = \begin{bmatrix} 1 & -1 & -2 & 1 \\ 3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 0 & 0 & -3 \\ 0 & 5 & 0 & 18 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 5 \end{bmatrix} \)
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\(P = \begin{bmatrix} 1 & -1 & 1 & 0 \\ -3 & 6 & 0 & 1 \\ -3 & 12 & 0 & 0 \\ 3 & 0 & 0 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 3 & 1 & \frac{20}{3} \\ 0 & 5 & -2 & -\frac{2}{3} \\ 0 & 0 & 5 & -\frac{1}{3} \\ 0 & 0 & 0 & 5 \end{bmatrix} \)
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\(P = \begin{bmatrix} 1 & 1 & 1 & 0 \\ -1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 0 & -6 & 16 \\ 0 & 5 & 5 & -13 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 5 \end{bmatrix} \)
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\(P = \begin{bmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & 2 & 0 \\ 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix} 5 & 0 & -1 & 3 \\ 0 & 5 & 4 & -\frac{32}{3} \\ 0 & 0 & 5 & \frac{2}{3} \\ 0 & 0 & 0 & 5 \end{bmatrix} \)
- \( c_A(\lambda) = (\lambda + 1)^3 \)
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\( \mathcal{B}_{E_{-1}(A)} = \left\{
\begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix},
\begin{bmatrix} 1 \\ 0 \\ 4 \end{bmatrix}
\right\} \)
Geometric multiplicity is not equal to algebraic multiplicity for \( \lambda = -1 \).
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\( \mathcal{B}_{E^2_{-1}(A)} = \left\{
\begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix},
\begin{bmatrix} 1 \\ 0 \\ 4 \end{bmatrix},
\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}
\right\} \)
\( G_{-1}(A) = E^2_{-1}(A) = \mathbb{R}^3 \)
- \( c_A(\lambda) = (\lambda + 1)^2 (x - 1)^2 \)
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\( \mathcal{B}_{E_{-1}(A)} = \left\{
\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}
\right\} \)
Geometric multiplicity is not equal to algebraic multiplicity for \( \lambda = -1 \).
\( \mathcal{B}_{E_{1}(A)} = \left\{
\begin{bmatrix} 5 \\ 2 \\ 2 \\ 1 \end{bmatrix}
\right\} \)
Geometric multiplicity is not equal to algebraic multiplicity for \( \lambda = 1 \).
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\( \mathcal{B}_{E^2_{-1}(A)} = \left\{
\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix},
\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}
\right\} \)
\( G_{-1}(A) = E^2_{-1}(A) \)
\( \mathcal{B}_{E_{1}(A)} = \left\{
\begin{bmatrix} 5 \\ 2 \\ 2 \\ 1 \end{bmatrix},
\begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \end{bmatrix}
\right\} \)
\( G_1(A) = E^2_1(A) \)