Question about Jordan form - Linear algebra Quick question,
We are given that the characteristic polynomial of a matrix $A$ is $P_A(x)=(x-1)^4$
We are asked to find all the possible jordan forms of $A$.
Obviously the minimal polynomial of $A$ can be $m_A(x)=(x-1)^k$, $ 1 \leq k \leq 4$
I understand that the size of the largest block in the jordan form is $k$, because The exponent of the term corresponding to an eigenvalue in the minimal polynomial is the size of the largest block.
But there is also a thing we can say about the characteristic polynomial, but I don't understand it.
We know that "The algebraic multiplicity of an eigenvalue is the sum of the total sizes of the blocks." Could someone please explain and demonstrate what this means?

The possibilities are
$$\begin{pmatrix}1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\end{pmatrix}\;\;,\;\;\begin{pmatrix}1&1&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\end{pmatrix}$$
$$\begin{pmatrix}1&1&0&0\\
0&1&0&0\\
0&0&1&1\\
0&0&0&1\end{pmatrix}\;\;,\;\;\begin{pmatrix}1&1&0&0\\
0&1&1&0\\
0&0&1&0\\
0&0&0&1\end{pmatrix}$$
$$\begin{pmatrix}1&1&0&0\\
0&1&1&0\\
0&0&1&1\\
0&0&0&1\end{pmatrix}$$
and the corresponding minimal pol. in each case is
$$(x-1)\;,\;\;(x-1)^2\;,\;\;(x-1)^2\;,\;\;(x-1)^3\;,\;\;(x-1)^4$$
Cases $\;2-3\;$ depend on whether the geometry multiplicity of the eigenvalue (i.e., the eigenspace's dimension) is two or one
