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I need some online tool for diagonalizing 2x2 matrices or at least finding the eigenvectors and eigenvalues of it. I don't like to download any stuf because I'm not able to, some online tool will do the job. Thanks.

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    $\begingroup$ Wolfram Alpha works. Try this. $\endgroup$ – J. M. is a poor mathematician Nov 4 '10 at 13:26
  • $\begingroup$ @Guesswhoitis. Of course, WolframAlpha pretty much can do everything, and it's free so it's awesome. $\endgroup$ – Derek 朕會功夫 May 10 '15 at 3:58
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Wolframalpha has an option. Try this: http://www.wolframalpha.com/examples/Matrices.html

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Let $A$ be your $2$ by $2$ diagonalizable matrix, let $\lambda$ and $\mu$ be its eigenvalues, and let $I$ be the $2$ by $2$ identity matrix.

If $\lambda=\mu$, then $A=\lambda I$ and there is nothing to do.

If $\lambda\not=\mu$, then the nonzero columns of $A-\mu I$ (such always exist) are $\lambda$-eigenvectors.

[Recall that the eigenvalues of $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$ are the roots of $X^2-(a+d)\,X+ad-bc$.]

[There is an obvious generalization to $n$ be $n$ matrices: in the above recipe to get a $\lambda$-eigenvector, replace $A-\mu I$ by the product of the $A-\mu I$, where $\mu$ runs over the eigenvalues not equal to $\lambda$.]

To prove this in the $2$ by $2$ case, it suffices to check $$A^2-(a+d)\,A+(ad-bc)\,I=0,$$ which is straightforward. This is (a particular case of) the Cayley-Hamilton Theorem.

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    $\begingroup$ The OP is asking for online tools. $\endgroup$ – Rasmus Aug 18 '11 at 10:56
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    $\begingroup$ Dear @Rasmus: Thanks. I agree. [The OP doesn’t seem to be around anyway, but this changes nothing.] I just wanted to make sure the (virtual) OP was aware that s/he was asking for online tools for solving a quadratic equation. I suspect that s/he was not aware of that. Some people believe that to find the eigenvectors in this case you must solve linear systems. [There are probably more online tools that solve quadratic equations than online tools that diagonalize matrices. This tells the OP which kind of online tools are needed.] ... But I'm open to dialogue... $\endgroup$ – Pierre-Yves Gaillard Aug 18 '11 at 11:26
  • $\begingroup$ I see your point now. Thank you for the explanation. +1 $\endgroup$ – Rasmus Aug 18 '11 at 14:42
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Try the Online Matrix Calculator.

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  • $\begingroup$ And which option does the $S \Lambda S^{-1}$ decomposition? I do not see it. $\endgroup$ – Val Jul 30 '13 at 12:07
  • $\begingroup$ @Val, check Eigenvalues/eigenvectors. $\endgroup$ – lhf Jul 30 '13 at 13:53
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    $\begingroup$ It produces you the list of eigenvalues/eigenvectors. It does not put them into the $S \Lambda S^{-1}$ form, which is what I call "diagonalization". matrixcalc.org/en.index.html does. $\endgroup$ – Val Jul 31 '13 at 10:31
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http://matrixcalc.org/en.index.html decomposes the matrix into $S \Lambda S^{-1}$

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  • $\begingroup$ It requires the eigenvalues to be rational, which is quite stupid. Otherwise it displays "Not Enough Rational Eigenvalues". $\endgroup$ – Alex M. Jan 12 '16 at 10:01

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