QR分解法(QR decomposition)

QR decomposition divides a m by n matrix $A$ into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$:

$$A=QR$$

Thus
$$Ax=b=>QRx=b=>Q^{T}QRx=Q^{T}b=>Rx=Q^{T}b$$
QR decomposition can be implemented by several algorithms, such as Gram–Schmidt process, Householder transformations, or Givens rotations. Gram-Schmidt procedure is a sequence of multiplications of A from the right by upper triangular matrices. Householde decomposition $A$ into $QR$ with orthogonal matrices. As orthogonal transformations are stable, using Householder triangularization and back-substitution to slove $Ax=b$ is backward stable[9][10].

##Householder transformation
Householder transformation reflects a vector $u$ about a hyperplane which orthogonal to a vector $v$ which is called Householder vector[8].
$$u^f=u-2v{v}^{T}u$$

The basic idea of Householder reflection for QR decomposition is to find a linear transformation that changes vector $u$ into a vector which collinear to $e_i$, $u^f=||u</$