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Is there any simple way to do this? The normal process of finding a diagonal matrix takes too long.

user322630
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So let's take two arbitrary matrices $\begin{pmatrix} a& b \\c& d\end{pmatrix}$ and $\begin{pmatrix} e& f \\g& h\end{pmatrix}$. You should know the following result:

If two matrices are similar, then they have the same eigenvalues.

The converse is not true, however the necessary condition is quite strong itself in the $2*2$ case. There are only two eigenvalues, they are determined by the trace of the matrix (sum of diagonal entries, and also of eigenvalues) and determinant of the matrix (product of eigenvalues).

In this case, that means at least that:

$a+d=e+h$, $ad-bc=eh-gf$. If one of these two is not satisfied, your matrices are not similar.

If the condition above is satisfied, it's still possible that the matrices are not similar. for example, $\begin{pmatrix} 1& 0 \\0& 1\end{pmatrix}$ and $\begin{pmatrix} 1& 2 \\0& 1\end{pmatrix}$ have the same eigenvalues, but are clearly not similar, because the identity is only similar to itself.

Your inspection stops here, unfortunately. This page: How do I tell if matrices are similar? will be of help to you from here, where the techniques used do not involve inspection, unfortunately.

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Sarvesh Ravichandran Iyer
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  • there is a reliable method in this video: https://www.youtube.com/watch?v=zrzMhU_4m-g – Will Jagy Apr 21 '16 at 04:33
  • "students should not be asking for clever tricks that avoid effort, they should be asking for methods that always work"-is that the reason? – Sarvesh Ravichandran Iyer Apr 21 '16 at 04:35
  • That's about what I had in mind. On the other hand, when I give answers or comments that ask them for specific effort, they almost never do it. Sorry about Aston Villa. – Will Jagy Apr 21 '16 at 04:39
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    Oh no, no problem. You are correct, however I encourage these people because I feel they are attempting to detach themselves from doing what they would usually do in front of a computer: type it into Wolfram or into Matlab, and get an answer. That's all. About Aston Villa, that's a different story, far grimmer than the one we have on this page. – Sarvesh Ravichandran Iyer Apr 21 '16 at 04:45
  • Something in Spain, top two tied on points and Real Madrid down just one. I suppose I do give shortcuts when I know such things; it does make me nervous when they demand something easy, as here saying the usual method takes too long. – Will Jagy Apr 21 '16 at 04:49
  • Spain is interesting, unlike France (certainly!). Besides, you are right again about them demanding easy stuff, it's some that a user warned me about recently, spoon feeding these newbies. As for me, well, I'm preparing for the Championship, so I'll see you around sometime. – Sarvesh Ravichandran Iyer Apr 21 '16 at 04:51