Why for random matrix one of eigenvalues is so big?

Question

Here i have the image of eigenvalues for matrixes obtained with $rand(n,n)/sqrt(n)$, why one of the eigenvalues for every matrix is so big on real axis comparatively to the others?

EDIT: i have several experiments for every matrix size, every time i have only one outstandingly big eigenvalue.

green: 8*8
blue: 16*16
red: 32*32
cyan: 64*64

EDIT: In case i use $randn$ instead of $rand$ while forming matrices, i have no outlier.

Observation: When matrix generated with rand, all elements are close to 1/2. So, you can fit vector to have biggest eigenvalue near $n/2.$ This is the same as in the case of Markov chains, matrix has eigenvalue 1 and some smaller eigenvalues.

Answer 1

If you are trying to verify the semicircular law, the elements need to be chosen from a Gaussian distribution, which is why randn has no outliers.

The outlier effect is a consequence of Perron-Frobenius theory; you are generating a non-negative matrix, so there is a single largest real eigenvalue, and all the others are strictly smaller. There is apparently a theorem about a similar scenario (symmetric real) described here: http://blogs.sas.com/content/iml/2012/05/16/the-curious-case-of-random-eigenvalues/