= Gaussian random matrix
{c}
{title2=$X_{ij}\overset{\mathrm{iid}}\sim N(0,1)$}
A standard real Gaussian <random matrix> has <independent and identically distributed> <standard normal> entries. Each column is an <orthogonally invariant random vector>: its <probability density function> is proportional to $\exp(-\|z\|_2^2/2)$. Consequently multiplying on either side by a fixed <orthogonal matrix> preserves the joint distribution. Its <singular values> are random; the expressions $\sqrt m\pm\sqrt n$ for an $m\times n$ <matrix> describe useful spectral scales, not deterministic endpoints that hold for every realization.
Back to article page
