QR Decomposition

Definition

$\mathbf{A}=\mathbf{QR}$ Decomposition of matrix $\mathbf{A}$ into a product $\mathbf{A}=\mathbf{QR}$ of an orthonormal matrix $\mathbf{Q}$ and an upper triangular matrix $\mathbf{R}$.

Computation

Using the Gram Schmidt Orthonormalization

QR decomposition can be computed by Gram Schmidt Orthonormalization. Where $Q=[q_1,\dots ,q_n]$ is the matrix of orthonormal column vectors obtained by the orthonormalization and $\mathbf{R}=\left[\begin{array}{ccccc}{\mathbf{q}}_1^{⊺}{\mathbf{a}}_1& {\mathbf{q}}_1^{⊺}{\mathbf{a}}_2& {\mathbf{q}}_1^{⊺}{\mathbf{a}}_3& \cdots & {\mathbf{q}}_1^{⊺}{\mathbf{a}}_n\\ 0& {\mathbf{q}}_2^{⊺}{\mathbf{a}}_2& {\mathbf{q}}_2^{⊺}{\mathbf{a}}_3& \cdots & {\mathbf{q}}_2^{⊺}{\mathbf{a}}_n\\ 0& 0& {\mathbf{q}}_3^{⊺}{\mathbf{a}}_3& \cdots & {\mathbf{q}}_3^{⊺}{\mathbf{a}}_n\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\mathbf{q}}_n^{⊺}{\mathbf{a}}_n\end{array}\right]$