I am starting to learn how to compute the Jordan form of matrix and would appreciate it if someone could explain how to solve (or the basics ideas of) the following practice problem.
Suppose that $M$ is a $4 \times 4$ matrix with coefficients in $\mathbb C$. List the possible Jordan canonical forms for $M$, under each of the following scenarios:
(a) The characteristic polynomial of $M$ is $x^2(x^2 − 1)$.
(b) The characteristic polynomial of $M$ is $(x^2 + 1)^2$.
(c) The characteristic polynomial of $M$ is $x^4$.
(d) The minimal polynomial of $M$ is $x^2$.
(e) The minimal polynomial of $M$ is $x^3$.
Now, I know that the minimal polynomial divides the characteristic polynomial and that the characteristic polynomials of the Jordan blocks are the elementary divisions. But I can't figure out how to determine the actual matrices. (I'm also aware that since $\mathbb C$ is algebraically closed, $M$ indeed has a Jordan basis.)
The multiplicity of an eigenvalue as a root of the characteristic polynomial is the number of occurrences of that eigenvalue on the diagonal. The size of the largest block is the multiplicity of that eigenvalue as a root of the minimal polynomial. Let's take d) for example. We know that all the blocks will correspond to the eigenvalue $0$. Moreover, the largest block has size two. Therefore there are two options modulo reordering of the blocks: either a single block of size two (and two blocks of size one) or two of blocks of size two:
$$\begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0& \\ & & & 0 \end{pmatrix} , \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & & \\ & & 0 & 1 \\ & & 0 & 0\end{pmatrix}.$$
Let's do a) too. Since we're only given the characteristic polynomial, we don't know what the size of the largest block is. The eigenvalues are: $\pm 1$ and $x = 0$, the latter with algebraic multiplicity two. So either there are two Jordan blocks corresponding to $0$ (each of size one), or a single one (of size two). The possibilities for the Jordan forms are thus
$$\begin{pmatrix} 0 & & & \\ & 0 & & \\ & & 1& \\ & & & -1 \end{pmatrix} , \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & & \\ & & 1 & \\ & & & -1\end{pmatrix}.$$