QR using Householder transformations

QR factorization process using Householder transformations:

1. Start from $A$
2. Apply a series of Householder reflections: $Q_{n-1}^T\cdots Q_1^{T}A=R$ to create zeros below the diagonal. $Q_i^T$ creates zeros below $a_{ii}$.
3. QR is obtained implicitly in the form: $A=Q_1\cdots Q_{n-1}R.$

The cost of applying each $Q_i^T$ is $O(n^2)$. So the total cost is $O(n^3)$.

If matrix $A$ is $m\times n$, the cost is $O(m{n}^2)$.

QR factorization, Householder transformation, Applying a Householder transformation