Is this matrix diagonalizable? Wolfram Alpha seems to contradict itself...

Question

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?

Answer 1

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}}