QR分解
a 1 = ⟨ e 1 , a 1 ⟩ e 1 a 2 = ⟨ e 1 , a 2 ⟩ e 1 + ⟨ e 2 , a 2 ⟩ e 2 a 3 = ⟨ e 1 , a 3 ⟩ e 1 + ⟨ e 2 , a 3 ⟩ e 2 + ⟨ e 3 , a 3 ⟩ e 3 ⋮ a k = ∑ j = 1 k ⟨ e j , a k ⟩ e j {\displaystyle {\begin{aligned}{\boldsymbol {a}}_{1}&=\langle {\boldsymbol {e}}_{1},{\boldsymbol {a}}_{1}\rangle {\boldsymbol {e}}_{1}\\{\boldsymbol {a}}_{2}&=\langle {\boldsymbol {e}}_{1},{\boldsymbol {a}}_{2}\rangle {\boldsymbol {e}}_{1}+\langle {\boldsymbol {e}}_{2},{\boldsymbol {a}}_{2}\rangle {\boldsymbol {e}}_{2}\\{\boldsymbol {a}}_{3}&=\langle {\boldsymbol {e}}_{1},{\boldsymbol {a}}_{3}\rangle {\boldsymbol {e}}_{1}+\langle {\boldsymbol {e}}_{2},{\boldsymbol {a}}_{3}\rangle {\boldsymbol {e}}_{2}+\langle {\boldsymbol {e}}_{3},{\boldsymbol {a}}_{3}\rangle {\boldsymbol {e}}_{3}\\&\vdots \\{\boldsymbol {a}}_{k}&=\sum _{j=1}^{k}\langle {\boldsymbol {e}}_{j},{\boldsymbol {a}}_{k}\rangle {\boldsymbol {e}}_{j}\end{aligned}}}
これは行列の形に書くことができ、