You need to know: Know: matrix with real entries, eigenvalues, probability, random variable, jointly independent random variables, standard normal distribution, notation for expectation, almost sure convergence, small o notation.
Background: For positive integer n, let be matrix whose entries are jointly independent real random variables which (i) have exponential decay, i.e., for some constants (independent of n) and all , and (ii) for and all , where Z is the random variable having standard normal distribution.
The Theorem: On 9th June 2012, Terence Tao and Van Vu submitted to arxiv a paper in which they proved that has real eigenvalues asymptotically almost surely.
Short context: The study of eigenvalues of large random matrices is one of the central themes in probability theory. The Theorem estimates how many of the eigenvalues are real. Earlier, this result was known to hold only for the special case when all has standard normal distribution. In fact, the Theorem is just one out of many corollaries of a much more general theorem, which, very roughly speaking, states that many properties of the eigenvalues of a random matrix with independent entries depend only on the first four moments of the entries. Unlike previous results in the literature, this theorem does not require matrix to be symmetric. It also does not require entries to be identically distributed.
Links: Free arxiv version of the original paper is here, journal version is here.
![E[\xi_{ij}^k]=E[Z^k]](https://s0.wp.com/latex.php?latex=E%5B%5Cxi_%7Bij%7D%5Ek%5D%3DE%5BZ%5Ek%5D&bg=ffffff&fg=000000&s=0&c=20201002)
Theorems of the 21st century 
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