QR factorization

The QR decomposition is the decomposition of $A$ into $A=QR$ where $Q$ is orthogonal and $R$ is upper triangular.

We will assume that $r_{ii}$ is non-negative.

Why QR?

1. $Q$ is orthogonal and conserves the 2-norm.
2. $R$ is triangular and easy to solve.

Example of application:

• Least-squares problem
• Find $x^{\ast }$ such that $x^{\ast }={\min }_x\Vert Ax-b{\Vert }_2$.
• We can use $A=QR$ for this.
• The solution satisfies $R{x}^{\ast }=Q^{T}b$.