Online tool for diagonalizing matrices?

Question

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.

Answer 1

Wolframalpha has an option. Try this: http://www.wolframalpha.com/examples/Matrices.html

Answer 2

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.

Answer 3

Try the Online Matrix Calculator.