What can be Jordan forms of matrix with charcterstic polynomial $(x-3)^2 (x-2)^4$ and minimal polynomial $(x-3)(x-2)^2$

Question

The following question was asked in my linear algebra quiz and I was unable to think which result I should use to solve it.

Question: Let A be a $6 \times 6$ matrix over $\mathbb{R}$ with chacterstic polynomial =$(x-3)^2 (x-2)^4$ and minimal polynomial = $(x-3)(x-2)^2$ .Then Jordan canonical form of A can be? I have to find all the matrix which are possible.

Algebraic multiplicity of 2 and 3 are 4,2 respectively and Geometric multiplicity of 2 and 3 are 2 and 1 respectively.

But i don't know which result should be used to move foreward. I calculated these in case they are useful.

Can you please shed some light on this?

Answer 1

If a factor $(x-\lambda)^k$ appears in the characteristic polynomial, then $k$ is total size of all Jordan $\lambda$-blocks.

If a factor $(x-\lambda)^k$ appears in the minimal polynomial, then $k$ is the size of the largest Jordan $\lambda$-block.

In our case we get:

• size of the largest $3$-block is $1$ and the total size is $2$ so there are two $3$-blocks of size $1$
• size of the largest $2$-block is $2$ and the total size is $4$ we can have two $2$-blocks of size $2$ or one $2$-block of size $2$ and two $2$-blocks of size $1$

$$\begin{bmatrix} 3 & & & & & \\ & 3 & & & & \\ & & 2 & 1 & & \\ & & & 2 & & \\ & & & & 2 & 1 \\ & & & & & 2 \end{bmatrix}\quad\text{ or } \quad \begin{bmatrix} 3 & & & & & \\ & 3 & & & & \\ & & 2 & 1 & & \\ & & & 2 & & \\ & & & & 2 & \\ & & & & & 2 \end{bmatrix}$$

Answer 2

By assumption, $A$'s group of elementary divisors can be one of the following:

1. $\lambda - 3, \lambda - 3, (\lambda - 2)^2, (\lambda - 2)^2$;
2. $\lambda - 3, \lambda - 3, (\lambda - 2)^2, \lambda - 2, \lambda - 2$.

Therefore, up to row/column permutations, there are two possibilities of $A$'s Jordan form $\newcommand{\diag}{\mathrm{diag}}$ \begin{align*} & \diag(3, 3, J_{(2)}(2), J_{(2)}(2)), \\ & \diag(3, 3, J_{(2)}(2), 2, 2), \end{align*} where $J_{(m)}(\lambda)$ denotes the Jordan block of size $m$ associated with eigenvalue $2$.