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This question already has an answer here:

If $A$ is a $2 \times 2$ matrix with complex entries, then $A$ is similar over C to a matrix of one of the two types

$$ \left[ {\begin{array}{cc} a & 0 \\ 0 & b \\ \end{array} } \right] $$

$$ \left[ {\begin{array}{cc} a & 0 \\ 1 & a \\ \end{array} } \right] $$

When For $2 \times 2$ matrix $N$ such that $N^{2} = 0$, $N$ is similar to $$ \left[ {\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} } \right] $$

The source of the problem is by Hoffman and Kunze.

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marked as duplicate by leo, Pedro Tamaroff, user147263, Omnomnomnom, M. Vinay Sep 27 '14 at 1:44

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Do you mean over complex numbers $\mathbb{C}$? And by that you mean a matrix with entries in $\mathbb{C}$? Could you please show some of your work? $\endgroup$ – Secret Math Oct 14 '13 at 17:14
  • $\begingroup$ are you aware of Jordan canonical forms... ?? $\endgroup$ – user87543 Oct 14 '13 at 18:05
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Hint: What are the eigenvalues of $N$? There are two cases to consider, depending on how many linearly independent eigenvectors $N$ has.

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Hint :

As $A\in M_2(\mathbb{C})$ , corresponding polynomial $\det(A-xI)$ is a quadratic.

Now, possibilities for roots $r_1,r_2$ of $\det(A-xI)$ are :

  • $r_1=r_2\neq 0$
  • $r_1\neq r_2$
  • $r_1=r_2=0$

Do you see which case implies $N^2=0$???? what would be its corresponding matrix?

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  • $\begingroup$ No, My class haven't covered Nilpotency yet. We only covered Determinent function and Characteristic function $\endgroup$ – Block Jeong Oct 15 '13 at 12:35
  • $\begingroup$ by nilpotency, i mean some positive power of a matrix is zero matrix.... $\endgroup$ – user87543 Oct 15 '13 at 17:27

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