Error-contrast matrix for a full-rank fixed design

Let $X\in {\mathbb{R}}^{n\times p}$ have full column rank, with $p<n$, and let $\mathrm{col}(X{)}^{\perp }$ denote the orthogonal complement of its column space in Euclidean ${\mathbb{R}}^n$ (The orthogonal complement $W^{\perp }=\{v:⟨v,w⟩=0\text{ for all }w\in W\}$). An error-contrast matrix for $X$ is a matrix $K\in {\mathbb{R}}^{n\times (n-p)}$ whose columns form a basis of $\mathrm{col}(X{)}^{\perp }$. Thus $K$ has full column rank and $X^{T}K=0$, so $K^{T}Y$ is an $(n-p)$-dimensional vector of linear contrasts that annihilate $X\beta$. Such a matrix exists by A full-rank error-contrast matrix exists when n exceeds p ↗.

This column-basis convention makes the product $K^{T}Y$ dimensionally defined. The Gaussian restricted-likelihood definition below fixes one such $K$ independently of the covariance parameter.

The condition $p<n$ is needed for a nonzero error-contrast space. When $p=n$, the orthogonal complement has dimension zero and this page makes no variance-likelihood claim from contrasts.