Square independent-component model with standardized sources

Let $p\ge 1$ and let $X,S$ be random vectors in ${\mathbb{R}}^p$, considered through their probability laws. A square independent-component representation of $X$ is a representation in distribution

where $A\in G{L}_p(\mathbb{R})$ is an invertible mixing matrix and $S=(S_1,\dots ,S_p{)}^T$ has mutually independent real-valued components. Mutual independence means that for all Borel sets $B_1,\dots ,B_p\subseteq \mathbb{R}$,

$\mathbb{P}(S_1\in B_1,\dots ,S_p\in B_p)={\prod }_{j=1}^p\mathbb{P}(S_j\in B_j)\mathrm{.}$

Each source is standardized: $\mathbb{E}{S}_j=0$ and $\mathbb{E}{S}_j^2=1$. A source is Gaussian if its law has density $(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}$, and is non-Gaussian otherwise. All vectors are columns and matrix transposes use the standard Euclidean inner product. The definition specifies a latent representation, not a claim that it can be recovered uniquely from the law of $X$.

Hastie, Tibshirani, and Friedman, §14.7.2

Hyvärinen, Karhunen, and Oja, §7.2.2