Is there a relationship between the size of a matrix and the number of eigenvalues

Question

I'm trying to find out whether there is a relationship between the size of a matrix and the probability of all eigenvalues being negative. I would've thought that if larger matrices are more likely to have more eigenvalues (which makes sense logically) then there is more likely to be at least on positive eigenvalue in there somewhere. I am just unsure of how I can show that larger matrices have more/are likely to have more eigenvalues.

Answer 1

Arguably the starting point for questions like this is the classic paper

E.P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958) 325–327.

A more modern paper with a title that suggests it might interest you is this one:

On the singular values of Gaussian random matrices, Jianhong Shen, Linear Algebra and its Applications 326 (2001) 1–14. See https://pdfs.semanticscholar.org/ba9d/7daa0456216335d22c58135e5a5b18b93a8b.pdf

It at least contains references to recent work on the topic.

I found both by a google search on ""on the distribution of characteristic values of random matrices", which I had mistakenly remembered as the title of Wigner's paper. Perhaps a more targeted google search can help you, as well.