- #1
Kamekui
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Homework Statement
1. Homework Statement [/b]
Enumerate all possible Jordan forms for 3 x 3 systems where all the eigen-values have negative real parts. Do not use specific values. Instead, use possibilities
like λ1; λ2; λ3, each with multiplicity 1, or λ (multiplicity 3).
Homework Equations
The Attempt at a Solution
Let Ji be the Jordan Form
J1=\begin{bmatrix}
λ1 & 0 & 0 \\
0 & λ2 & 0\\
0 & 0 & λ3
\end{bmatrix}
So λ1, λ2, and λ3 all have multiplicity 1
J2=\begin{bmatrix}
λ1 & 0 & 0 \\
0 & λ2 & 1\\
0 & 0 & λ2
\end{bmatrix}
λ1 (Multiplicity 1), λ2 (Multiplicity 2)
J3=\begin{bmatrix}
λ1 & 0 & 0\\
0 & λ1 & 0\\
0 & 0 & λ1
\end{bmatrix}
λ1 (Multiplicity 3) With 1 generalized eigenvector
J4=\begin{bmatrix}
λ1 & 1 & 0\\
0 & λ1 & 1\\
0 & 0 & λ1
\end{bmatrix}
λ1 (Mulitiplicity 3) With 2 generalized eigenvectors
J5=\begin{bmatrix}
λ1 & 0 & 0 \\
0 & λ2 & 0\\
0 & 0 & λ3
\end{bmatrix}
Where λ1 ε ℝ, λ2 and λ3 are complex conjugates such that
λ2= -a+bi and λ3=-a-bi. So λ1, λ2, and λ3 all have multiplicity 1.
1) Do these Jordan Forms look correct?
2) Are there more? ( I think there may be 3 more but I'm unsure)