Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant, trace and characteristic polynomial, but that are not similar to each other.
I come up with two matrices:
$A=\begin{pmatrix}
0 &1 \\
0 &0
\end{pmatrix}$ and $B=\begin{pmatrix}
0 &0 \\
1 &0
\end{pmatrix}.$
It is easy to check that they have same rank, determinant, trace and characteristic polynomial. However, my question is I do not know how to prove two matrices are similar or not.
I have learnt the converse in my textbook, i.e. If two matrices are similar, they have the same determinant, characteristic polynomial,etc.
I have also known that (but I do now know the proof), we can check by using Jordan form of two matrices. I do not know if this claim is correct:
"If two matrices have the same Jordan form, they are similar to each other."
Yet, go back to the question, is it a quick way to prove? Thank you in advance.
Best Answer
Try $A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$. They are both rank 2.