# Jordan Forms for a 3x3 Matrix

1. Mar 12, 2012

### Kamekui

1. The problem statement, all variables and given/known data
1. The problem statement, all variables and given/known data[/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).

2. Relevant equations

3. 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)

2. Mar 13, 2012

### HallsofIvy

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

3. Mar 13, 2012

### Kamekui

Sorry if I seem confused but what blank spaces?