<s>[INST] Determine if matrices A and B are similar Determine if matrices $A$ and $B$ are similar.
$A=\begin{pmatrix}1 & 1 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 3 & 0 \\0 & 0 & 0 & 3\end{pmatrix}\: $
and
$B=\begin{pmatrix}1 & 1 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 3 & 1 \\0 & 0 & 0 & 3\end{pmatrix}$
I found the characteristic polynomials to be the same for both matrices
$(1-λ)(3-λ)(3-λ)(1-λ)$
so I went to plug in the values to find the eigenvectors for matrix $A$ and I think I may have screwed up because 
$\begin{pmatrix}0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 2 & 0 \\0 & 0 & 0 & 2\end{pmatrix}$
I got an eigenvector for $λ=1$ to be $\begin{pmatrix}0 \\0 \\0 \\0\end{pmatrix}$ which I'm not even sure if I did it correctly or not because I don't think this is possible.
for $λ=3$ I got $\:\begin{pmatrix}1 \\\frac{1}{2} \\0 \\0\end{pmatrix}$
If I did do this correctly then is the geometric multiplicity for both = 1?
 [/INST] Both matrices are in Jordan normal form. This normal form is unique for all similar matrices. Henece $A$  and $B$ are not similar.
Besides, these Jordan normal forms tell us the eigenspace $E_1$ (for the eigenvalue $1$ has dimension $1$ for  both matrices, while the eigenspace $E_3$ has dimension $2$ for $A$ and $1$ for $B$.
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