QR Factorization

Definition

A matrix $A$ that can be represented as a Matrix Multiplication of $A=QR$ where:

• $Q$ is Unitary
• $R$ is Upper Triangular When $A$ is full rank,
• $A=[a_1,\dots ,a_n]$
• $\{e_1,\dots ,e_n\}$ is the Orthonormal Basis constructed from Gram-Schmidt Algorithm of $\{a_1,\dots ,a_n\}$
• $Q=[e_1,\dots ,e_n]$

\langle a_1 | e_{1} \rangle & \dots & & \langle a_{n} | e_{1} \rangle \ 0 & \langle a_{2} | e_{2} \rangle & \dots & \langle a_{n} | e_{2} \rangle \ \vdots\ 0 & \dots & & \langle a_{n} | e_{n} \rangle \end{array}\right]$$