QR Factorization

For any $m\times n$ matrix $A$, with linearly independent columns:

$$A=QR$$

Q is

• m x n
• its columns are an orthonormal basis for Col A R is
• n x n
• upper triangular
• positive diagonal
• $||{\vec{r}}_j||=||{\vec{a}}_j||$ where $\vec{r}$ and $\vec{a}$ are columns of R and A