# Similar matrices - find a matrix $T$

The matrices $A=\begin{pmatrix}5 & -3 \\ 4 & -2\end{pmatrix}$ and $B=\begin{pmatrix}-1 & 1\\-6 & 4\end{pmatrix}$ are similar. By knowing that similar matrices have the same eigenvalues, find a matrix $T$ such that $A=TBT^{-1}.$

any idea or proof is welcome :) thanks .

• No, they are not similar. – Chris Eagle Dec 2 '12 at 16:55
• They cannot be similar because neither their determinant nor their trace are equal...unless you're working on a field of characteristic $\,2\,$ ... – DonAntonio Dec 2 '12 at 16:55
• sorry, I fix the mistake – Iuli Dec 2 '12 at 16:56
• Oh, now that looks better. – DonAntonio Dec 2 '12 at 16:58

## 1 Answer

Evaluate $\,A'$s eigenvalues:

$$p_A(t):=\det(tI-A)=\left|\begin{array}{}t-5&\;\;\;3\\-4&t+2\end{array}\right|=t^2-3t+2=(t-2)(t-1)$$

Thus, the eigenvalues of $\,A\,$ are $\,1,2\,$. Find now one eigenvector for each eigenvalue:

$$(i)\;\;t=1:\;\;\;\;\;\;-4x+3y=0\Longleftrightarrow y=\frac{4}{3}x\Longrightarrow \binom{3}{4}$$ $${}$$

$$(i)\;\;t=2:\,\,\,\,\,\,-3x+3y=0\Longleftrightarrow x=y\Longrightarrow \binom{1}{1}$$

Well, as we know, we get that

$$S=\left(\begin{array}{}3&1\\4&1\end{array}\right)$$

Take it from here

• But $A \neq TBT^{-1}$ – Iuli Dec 2 '12 at 17:12
• I messed up both the letter and the rows and columns of the last matrix. Now, $$\,S^{-1}AS=\begin{pmatrix}1&0\\0&2\end{pmatrix}=R^{-1}BR\,$$ ,and you can get the matrix $\,R\,$ for $\,B\,$ by the same method as we got $\,S\,$ for $\,A\,$ above, so now yes: take it from here – DonAntonio Dec 2 '12 at 17:20
• Could you give me more details, please. After I found matrix $S$ which is the next step? Finding matrix $R$ and then what about $T$ ? thanks:) – Iuli Dec 2 '12 at 17:21
• Read my last, edited message, @Iuli – DonAntonio Dec 2 '12 at 17:22
• @DonAntonie Thanks :) it is interesting. Could you give some ideas about math.stackexchange.com/questions/247720/…. merci :) – Iuli Dec 2 '12 at 17:27