QR Factorization

Orthonormal Basis

If vectors have unit lengths they are orthonormal.

To find the orthonormal basis of $\vec{v}$, $\hat{v}$

$$\hat{v}=\frac{1}{||\vec{v}||}\vec{v}$$

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QR Factorization

For a $m\times n$ matrix $A$ linearly independent columns,

$$A=QR$$

$Q$ is an $m\times n$, with columns are an orthonormal basis for $\text{Col}A$. $R$ is $n\times n$, upper triangular, with positive entries on its diagonal.

We can get $Q$ using the Gram-Schmidt process. To find $R$, we can us $R=Q^{T}A$ due to $Q^{T}Q=I$

Properties

$\text{ Length of the }{j}^{th}\text{ column of }R=\text{ length of the }{j}^{th}\text{ column of }A$