# Linear Algebra : Check similarity [duplicate]

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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.

## marked as duplicate by leo, Pedro Tamaroff♦, user147263, Omnomnomnom, M. VinaySep 27 '14 at 1:44

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• 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? – Secret Math Oct 14 '13 at 17:14
• are you aware of Jordan canonical forms... ?? – user87543 Oct 14 '13 at 18:05

## 2 Answers

Hint: What are the eigenvalues of $N$? There are two cases to consider, depending on how many linearly independent eigenvectors $N$ has.

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?

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