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.
Online tool for diagonalizing matrices?
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13Wolfram Alpha works. Try [this](http://tinyurl.com/2vtv897). – 2010-11-04
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0@Guesswhoitis. Of course, WolframAlpha pretty much can do everything, and it's free so it's awesome. – 2015-05-10
4 Answers
Wolframalpha has an option. Try this: http://www.wolframalpha.com/examples/Matrices.html
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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1The OP is asking for online tools. – 2011-08-18
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2Dear @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... – 2011-08-18
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0I see your point now. Thank you for the explanation. +1 – 2011-08-18
Try the Online Matrix Calculator.
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0And which option does the $S \Lambda S^{-1}$ decomposition? I do not see it. – 2013-07-30
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0@Val, check Eigenvalues/eigenvectors. – 2013-07-30
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1It 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". http://matrixcalc.org/en.index.html does. – 2013-07-31
http://matrixcalc.org/en.index.html decomposes the matrix into $S \Lambda S^{-1}$
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0It requires the eigenvalues to be rational, which is quite stupid. Otherwise it displays "Not Enough Rational Eigenvalues". – 2016-01-12