TITLE: Can a matrix be invertible but not diagonalizable?
QUESTION [25 upvotes]: While reading a chapter on diagonalizable matrices, I found myself wondering:

Can a matrix $A \in \mathbb R^{n \times n}$ be invertible but not diagonalizable?

My quick Google search did not return a clear answer.

REPLY [17 votes]: Geometric(-ish) answer. Take, in $\mathbb R^2$, a rotation of angle $0<\theta<2\pi$ with $\theta \neq \pi$. Then the associated matrix is invertible (the inverse being the rotation of $-\theta$) but is not diagonalisable, since no non-zero vector is mapped into a multiple of itself by a rotation of such angles.