# Is this matrix diagonalizable? Wolfram Alpha seems to contradict itself…

I have the matrix $\begin{bmatrix}0.45 & 0.40 \\ 0.55 & 0.60 \end{bmatrix}$.

I believe $\begin{bmatrix}\frac{10}{17} \\ \frac{55}{68}\end{bmatrix}$ is an eigenvector for this matrix corresponding to the eigenvalue $1$, and that $\begin{bmatrix}-\sqrt{2} \\ \sqrt{2}\end{bmatrix}$ is an eigenvector for this matrix corresponding to the eigenvalue $0.05$.

However, Wolfram Alpha tells me this matrix is, in fact, not diagonalizable (a.k.a. "defective"):

I'm really confused... which one is in fact defective -- Wolfram Alpha, or the matrix?
Or is it my understanding of diagonalizability that's, uh, defective?

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I think you are correct. –  Daryl Oct 30 '12 at 2:53
@Daryl: By "correct" do you mean it is, in fact, diagonalizable? –  Mehrdad Oct 30 '12 at 2:54
Yes. I get the characteristic polynomial as $\lambda^2-1.05\lambda+0.05$, which has two distinct roots. Thus, for a $2\times2$ matrix, it is diagonalisable. –  Daryl Oct 30 '12 at 2:56
I think the "Result" part of WA is not highlighted thus, perhaps, that means it isn't appliable. Since WA itself is giving the Jordan form and this is diagonal, if the "Result" part applies then this is a huge contradiction. –  DonAntonio Oct 30 '12 at 2:56
@DonAntonio, Daryl: Yeah I didn't know what Jordan meant so I was a little confused... thanks for the info. –  Mehrdad Oct 30 '12 at 3:04

I agree with all the comments. Namely,

1. The matrix is clearly diagonalizable,
2. The rationalized version works correctly, and
3. Numerical linear algebra can be tricky and surprising.

In spite of points 2 and 3, I'd still call this a bug. Alpha is intended to guess the users intent. While clearly very hard, I don't think that interpreting numbers like $0.55$ as $55/100$ is too far out there. Even failing that, a small perturbation of the elements of the matrix don't change the fact that the matrix is diagonalizable.

Fortunately, there is an easy work around. Just enter:

diagonalize rationalize {{0.45,0.4},{0.55,0.6}}


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Cool trick with rationalize, thanks! –  Mehrdad Oct 30 '12 at 6:53