Possible Jordan Forms for 3x3 Matrix with Negative Eigenvalues

[Kamekui]

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 J_i 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)

 

[HallsofIvy]

Why the blank spaces? Were those supposed to be "1"s?

 

[Kamekui]

Sorry if I seem confused but what blank spaces?