Let $A$ be a $4×4$ matrix with complex entries such that $\sigma(A) = \{1, 2\}$. What are all possible Jordan forms of $A$?

Question

I calculated 12 Jordan forms thus far: $\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$, $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$.

Unsure if these are correct, or if there are more. Any explanation would be greatly appreciated!

Answer 1

The number $4$ has the following partition $\{ (4), (3,1), (2,2), (2,1,1), (1,1,1,1) \}$.

Let me use the notation $J_{\lambda, n}$ where $n$ denotes the size and $\lambda$ denotes the eigenvalue. $\lambda$ takes value $1, 2$ and $n$ takes value $1,2,3$.

Using multi-set notation

For partition $(3,1)$:

1. $\{J_{1,3}, J_{2,1}\}$,
2. $\{J_{2,3}, J_{1,1}\}$,

For partition $(2,2)$:

3. $\{ J_{2,2}, J_{1,2}\}$

For partition $(2,1,1)$:

4. $\{J_{1,2}, J_{1,1}, J_{2,1}\}$ [remark: you second and third matrices are from here].
5. $\{J_{1,2}, J_{2,1}, J_{2,1}\}$
6. $\{J_{2,2},J_{2,1}, J_{1,1} \}$ [remark: you second last and third last matrices are from here].
7. $\{J_{2,2}, J_{1,1}, J_{1,1}\}$

For partition $(1,1,1,1)$.

8. $\{J_{2,1}, J_{1,1},J_{1,1},J_{1,1} \}$
9. $\{J_{2,1}, J_{2,1},J_{1,1},J_{1,1} \}$
10. $\{J_{2,1}, J_{2,1},J_{2,1},J_{1,1} \}$

All these and depending on your convention, you might want to consider the permutation of the blocks.