Random matrix

Remarks

Equivalently, $X = (X_{ij})$ where each entry $X_{ij}$ is a random variable on the same space $(\Omega, \mathcal{F}, P)$. The law of $X$ is the probability measure $P \circ X^{-1}$ on $\mathbb{K}^{m \times n}$. Example: let $G$ be an $n \times n$ matrix whose entries are independent standard normal random variables, and let $H = (G + G^T)/\sqrt{2}$. Each entry of $H$ is a sum of measurable functions, so $H$ is a random symmetric matrix; it is a Wigner matrix with Gaussian entries. Non-example: $n^2$ random variables defined on different probability spaces do not form a random matrix, because the entries need one common probability space.