Full, reduced, and column-pivoted computational QR factorisations

Let $A\in M_{m\times n}(\mathbb{F})$ with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ and $m\ge n$.

A full QR factorisation of $A$ is a factorisation

$A=QR,$where $Q\in M_m(\mathbb{F})$ is orthogonal or unitary and $R\in M_{m\times n}(\mathbb{F})$ is upper trapezoidal. A reduced QR factorisation is a factorisation$A=\widehat{Q}\widehat{R},$where $\widehat{Q}\in M_{m\times n}(\mathbb{F})$ has orthonormal columns (${\widehat{Q}}^{\ast }\widehat{Q}=I_n$) and $\widehat{R}\in M_n(\mathbb{F})$ is upper triangular. A column-pivoted QR factorisation is a factorisation$A\mathrm{\Pi }=\widehat{Q}R,$

where $\mathrm{\Pi }$ is a permutation matrix acting on columns, $\widehat{Q}$ has orthonormal columns, and $R$ is upper trapezoidal or upper triangular according to whether the full or reduced form is used. Computationally, one obtains these factorisations by successive Householder reflectors or Givens transformations that zero entries below the diagonal, with the pivoted version choosing the next column after a permutation.