the Wigner matrix
/ Wigner = VIG-ner /
A Wigner matrix is the natural general random Hermitian matrix: symmetric (or Hermitian), with entries that are independent and identically distributed apart from the symmetry constraint, but whose entry distribution is otherwise arbitrary. Where the Gaussian ensembles demand Gaussian entries, the Wigner matrix asks only for independence, mean zero, and a finite variance. It is the object on which universality is tested: the claim that the spectral behaviour does not care whether the entries are Gaussian, Bernoulli, or anything else with the right first two moments.
Concretely, take an N-by-N real symmetric (or complex Hermitian) matrix W_N. The entries on and above the diagonal are independent. The off-diagonal entries are iid with mean 0 and variance sigma^2 (and the diagonal entries iid, often with a possibly different variance that does not affect the limit). To keep the spectrum bounded one normalises: set H_N = W_N / sqrt(N), so the off-diagonal entries have variance sigma^2/N. With this normalisation the typical eigenvalue is O(1) rather than growing with N, the row sums of squared entries are O(1), and the empirical spectral distribution of H_N converges to Wigner's semicircle law on [-2 sigma, 2 sigma]. The point of allowing arbitrary entry distributions (subject to moment conditions) is that this convergence holds regardless: it is the prototypical universality statement.
The Wigner matrix is the workhorse of the field because it is both general enough to be realistic and structured enough to analyse. Two complementary proof techniques apply to it: the moment method, which computes E[trace(H_N^k)] = E[sum lambda_i^k] by expanding into a sum over closed walks on indices and counting which walks survive (the non-crossing pairings), and the Stieltjes-transform/resolvent method, which studies (H_N - z)^(-1) and derives a self-consistent equation for its limit. A caveat: 'Wigner matrix' refers to the symmetric/Hermitian iid model. A matrix with iid entries and no symmetry imposed is a different object (its eigenvalues are complex and obey the circular law, not the semicircle), and a sample covariance matrix has correlated entries by construction — both are studied separately.
Build W_N by filling the upper triangle with independent +1/-1 fair-coin flips (Bernoulli, decidedly non-Gaussian), reflecting to make it symmetric, and dividing by sqrt(N). Its eigenvalue histogram still converges to the very same semicircle on [-2, 2] you would get from Gaussian entries. The entry distribution has vanished from the answer — that is universality in its simplest form.
Different entry laws, identical semicircle: the Wigner matrix is universality made concrete.
The sqrt(N) normalisation is not cosmetic: without it the typical eigenvalue grows like sqrt(N) and there is no limiting law. And the semicircle limit needs the variance of the off-diagonal entries to be finite; heavy-tailed entries (infinite variance) give a different, non-semicircular limit.