The measure of real matrices that are not diagonalizable
over $\mathbb{C}$ equals to 0, see for example <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3142770">On the computation of Jordan canonical form</A>, so the probability for a random matrix with a continuous probability distribution to be non-diagonalizable vanishes. To put it differently, the set of real matrices without multiple eigenvalues is dense, and a matrix without multiple eigenvalues is definitely diagonalizable.