Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a good reference for diagonalizability of random matrices?



Diagonalizability of Gaussian random matrices
The measure of real matrices that are not diagonalizable
over $\mathbb{C}$ equals to 0, see for example [On the computation of Jordan canonical form](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3142770), 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.


