Proving the Diagonalizability of a Real 2x2 Matrix Using Invertible Matrices

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Homework Statement

Let A be a 2x2 real matrix which cannot be diagonalized by any matrix P (real or complex). Prove there is an invertible real 2x2 matrix P such that

$$P^{-1}AP = \left( \begin{array}{cc} \lambda & 1 \\ 0 & \lambda \end{array} \right)$$

I know how to diagonalize a matrix by using eigenvectors but I don't think that really helps here. I tried proving it by letting A be {a, b, c, d} and P be {e, f, g, h} but it gets really messy and I don't think that's the right way to do it. Any help appreciated!

 

[HallsofIvy]

Well, it does help a little. It means you know that if a 2 by 2 matrix has two independent eigenvectors, then it can be diagonalized. And, of course, if the matrix has two distinct eigenvalues, then their eigenvectors are independent. Here, your matrix must have only one eigenvalue (which may be complex) and only one eigenvector. You might try this: choose your basis so that one of the basis vectors is that eigenvector and the other is orthogonal to it.

 

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Thanks. So do you mean: view the transformation associated with matrix A in a basis of {eigenvector, orthogonal to eigenvector} and find the matrix for the transformation in this basis?
I'm not sure what the significance of the orthogonal vector is here.