Franklin's Notes

QR Factorization

Given a matrix $A\in\mathbb{C}^{m\times n}$ with columns $a_1,...,a_n$, the columns spaces of successive columns of $A$ are nested: and the idea behind QR factorization is to construct orthonormal bases for these successive column spaces. In particular, we seek orthonormal vectors $q_1,...,q_n$ such that for each $j$. Given such a set of vectors ${q_i}$, we would have that each $a_k$ is a linear combination of the vectors $q_1,...,q_k$, or for some coefficients $r_{ik}$. This in turn implies that where $Q=[q_1|...|q_n]$ and $R$ is upper-triangular matrix with entries $(r_{ij})$. A factorization $A=QR$, where $Q$ is unitary and $R$ is upper-triangular, is called a QR factorization.

If $A$ is not a square matrix or is not full rank, then the matrix $Q$ constructed in this way way not actually be unitary, because it may not even be square. A factorization $A=\hat{Q}\hat{R}$ is called a reduced QR factorization if $Q$ has orthonormal columns whose span is the columns space of $A$, but is not a square matrix. A full QR factorization takes the form $A=QR$ with $Q$ unitary, and in the case where $A$ is not square or not full rank, $R$ may be rectangular and/or have some "silent rows" that obliterate the extra columns of $Q$ that do not lie in the column space of $A$.