- Assume that we have a symmetric random matrix M where the entries Mij=Mji are each drawn from some probability distribution pij. Furthermore assume that pij(x)=pij(−x), i.e., that the distribution is symmetric (see e.g., [Wigner, 1958] for details).
2.1. Prove that the distribution over eigenvalues is also symmetric. That is, for any eigenvector v the probability that the associated eigenvalue λ satisfies P(λ>0)=P(λ<0).
2.2. Why does the above not imply P(λ>0)=0.5?
I am attempting 2.2. What is the probability that the eigenvalue is 0, that is, P(λ=0)? Eigenvalue is 0 for a singular matrix. The 2021 paper titled “The singularity probability of a random symmetric matrix is exponentially small” settles the question when pij is such that the elements are independently and uniformly selected.
Therefore, the question seems to imply that there are other distributions pij for which the probability of singularity is non-trivial.