Gaussian random matrix ($X_{ij}\stackrel{\mathrm{iid}}{\sim }N(0,1)$)

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 $\exp (-\Vert z{\Vert }_2^2/2)$. 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.