Given nullities, how would I determine the Jordan form of a matrix $A$?

Question

Given a $12 \times 12$ matrix with only one eigenvalue $\lambda = 7$, if we're given that
$nullity (A-7I) = 4, nullity (A-7I)^2 = 7, nullity (A-7I)^3 = 10, nullity (A-7I)^4 = 12$ how would I find a Jordan form of $A$? I don't understand the solution at all, they say the jordan form will consist of

$4$ blocks, with sizes $4,4,3,1$. How would you find this?

Answer 1

For an eigenvalue $\lambda$, the nullity of $A-\lambda I$ is the number of Jordan blocks.

The nullity of $(A-\lambda I)^2$ is the number of Jordan blocks plus the number of Jordan blocks of size at least $2$.

The nullity of $(A-\lambda I)^3$ is the number of Jordan blocks plus the number of Jordan blocks of size at least $2$ plus the number of Jordan blocks of size at least $3$. Etc.

In your example you have $4$ Jordan blocks. Then $7$ is $4$ plus the number of Jordan blocks of size $\ge2$. So there are three of those and so exactly one of size $1$.

Then $10$ is $4+3$ plus the number of Jordan blocks of size $\ge3$. So all three Jordan blocks of size $\ge2$ actually are of size $\ge3$. There are no size $2$ Jordan blocks. Etc.