# QR Factorization Any matrix $A \in \mathbb{C}^{m \times n}$ with $m \geq n$ has a QR factorization $A=Q R$, where $Q \in \mathbb{C}^{m \times m}$ is unitary and $R$ is upper trapezoidal, that is, $R=\left[\begin{array}{c}R_1 \\ 0\end{array}\right]$, where $R_1 \in \mathbb{C}^{n \times n}$ is upper triangular. Partitioning $Q=\left[Q_1 Q_2\right]$, where $Q_1 \in \mathbb{C}^{m \times n}$ has orthonormal columns, gives $A=Q_1 R_1$, which is the reduced, economy size, or thin $\mathrm{QR}$ factorization. --- ## References