The real-eigenvalue count conjecture for random sign matrices

The conjecture predicts that a random sign matrix has on the order of the square root of its size in real eigenvalues with high probability. The conjecture arose from a 2009 conversation involving P. M. Wood; the survey also records that Babai proposed it privately in the 1970s.

Known results

• For Gaussian matrices, the expected number of real eigenvalues is asymptotic to $\sqrt{2n/\pi}$ (Edelman, Kostlan, and Shub).
• Tao and Vu obtained asymptotic results for certain matrices with entries in $\{0,\pm1\}$, but not for Rademacher matrices.
• Even proving that a Rademacher matrix has at least $2$ real eigenvalues with high probability was recorded as open in 2016 and again in 2020.

Current status (as of August 2026): The $\Theta(\sqrt n)$ conjecture remains open, and even the weaker high-probability existence of at least $2$ real eigenvalues has no recorded proof here.