Size of Jordan block

Question

Imagine that I'm writing the Jordan form of a matrix and I know that the eigenvalue needs to appear 4 times in the diagonal (algebraic multiplicity is 4) and we need 2 Jordan blocks (geometric multiplicity is 2). Now how do I know the size of the blocks? It could be a 1x1 and 3x3 block ou two 2x2 blocks, right?

Answer 1

Yes, the two options you give are the only ones. To decide between the two you can consider $(A-\lambda I)^2$ where $A$ is the matrix and $\lambda$ the eigenvalue.

If the dimension of the kernel is $3$ then you are in the $(1,3)$ situation, if it is $4$ then you are in the $(2,2)$ situation.

More generally, the increase in dimension between the kernel of $(A-\lambda I)^{r-1}$ and $(A-\lambda I)^{r}$ is the number of blocks of size $r$ or more.

Answer 2

A. You have as many blocks, to an eigenvalue, as the geometric multiplicity. Hence, in your case two blocks, on cumulative size 4. Thus, either 2+2 or 3+1.