観測方程式の行列表示
Ｖ＝ＡＸ−Ｌ，Ｐ
ただし、
Ｖ：残差のベクトル
Ａ：係数の行列
Ｘ：未知数のベクトル
Ｌ：定数項のベクトル
Ｐ：重量の行列
V T P V = ( A X − L ) T P ( A X − L ) {\displaystyle \ V^{T}PV=(AX-L)^{T}P(AX-L)}
= ( X T A T − L T ) P ( A X − L ) {\displaystyle \ =(X^{T}A^{T}-L^{T})P(AX-L)}
= X T A T P A X − X T A T P L − L T P A X + L T P L {\displaystyle \ =X^{T}A^{T}PAX-X^{T}A^{T}PL-L^{T}PAX+L^{T}PL}
∂ V T P V ∂ X = ∂ ( X T A T P A X ) ∂ X − ∂ ( X T A T P L ) ∂ X − ∂ ( L T P A X ) ∂ X + ∂ ( L T P L ) ∂ X {\displaystyle {\frac {\partial V^{T}PV}{\partial X}}={\frac {\partial (X^{T}A^{T}PAX)}{\partial X}}-{\frac {\partial (X^{T}A^{T}PL)}{\partial X}}-{\frac {\partial (L^{T}PAX)}{\partial X}}+{\frac {\partial (L^{T}PL)}{\partial X}}}
= ( X T A T P A ) + ( A T P A ) X − L T P T A − L T P A {\displaystyle \ =(X^{T}A^{T}PA)+(A^{T}PA)X-L^{T}P^{T}A-L^{T}PA}
= A T P A X + A T P A X − L T P T A − L T P A {\displaystyle \ =A^{T}PAX+A^{T}PAX-L^{T}P^{T}A-L^{T}PA}
= 2 A T P A X − 2 L T P A {\displaystyle \ =2A^{T}PAX-2L^{T}PA}
X = ( A T P A ) − 1 ( L T P A ) {\displaystyle \ X=(A^{T}PA)^{-1}(L^{T}PA)}
