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Simple
Simple
2:53 AM
Classify up to similarity all complex 3×3 matrices $A$ such that $A^3= 2A^2−A$.
Given that $A^3=2A^2-A$, we define $f(x)=x^3-2x^2+x=x(x-1)^2$. As $f(A)=\mathbf{O}$, the minimal polynomial $p(x)$ of $A$ divides $f(x)$. So, the possible candidates for $p(x)$ are
$$x,\,x-1,\,x(x-1),\,(x-1)^2,\,x(x-1)^2$$
When $p(x)=x$, we have $A$ equals to the zero matrix which satisfies the condition.\\
When $p(x)=x-1$, we have $A$ equals to the identity matrix which satisfies the condition.\\
When $p(x)=(x-1)^2$, there is no such $A$ satisfies the given condition.\\
When $p(x)=x(x-1)$, the Jordan canonical form is
 
 
2 hours later…
Simple
Simple
5:11 AM
I got $J_1$ wrong
 
Martin Sleziak
Martin Sleziak
You claim that there is on matrix with minimal polynomial $p(x)=(x-1)^2$.
What about $J-I=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
$, i.e., $J=\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}$?
The matrix $\operatorname{diag}(0,1,1)$ gives another class of solutions for $p(x)=x(x-1)$.
@Simple I do not think it's wrong, but it is not the only class of similar matrices that fulfills this.
 
 
1 hour later…
Simple
Simple
6:48 AM
@MartinSleziak thank you, I totally missed that
 

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