### 0.1 Householder QR Decomposition


**Definition 0.1.1 **  The **Householder QR decomposition** of a matrix $A\in\mathbb{C}^{m\times n}$ is a set of unitary matrices $\{Q_n,\dots,Q_1\}\subset\mathbb{C}^{m\times m}$ such that the matrix $R\in\mathbb{C}^{m\times n}$ defined below is upper triangular.
\[R = Q_n\dots Q_1A\]

**Definition 0.1.2 **  The **Householder reflector** $H_v\in\mathbb{C}^{n\times n}$ for a unit vector $v\in\mathbb{C}^n$ is the following matrix.
\[H_v = I-2vv^*\]

**Proposition 0.1.3 **  For $x,y\in\mathbb{R}^n$ with $||x||=||y||$, and $x\neq y$, the Householder reflector $H_v$ for $v=\frac{x-y}{||x-y||}$ maps $x$ to $y$.
\[H_vx=y\]

**Proposition 0.1.4 **  Each $A\in\mathbb{R}^{m\times n}$ has a QR decomposition.

**Algorithm 0.1.5 **  The **Householder QR factorization algorithm** can be used to calculate the $QR$ decomposition for a matrix $A=QR$. The following algorithm computes $R$ by leaving the result in place of $A$ and then computes the columns of $Q$ by applying the Householder transformation for each of the standard basis vectors.