<s>[INST] 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?
 [/INST] 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
</s>