行列　ベクトル　微分 - daruma3940の日記

これであってるはずなのだけれど間違ってたら教えてほしいのじぇ。

$A = \left( \begin{array}{ccc} a_{11} & a_{21} \\ a_{12} & a_{22} \end{array} \right) \\ \vec{x }= \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) \\とすると\\ \vec{y}=A \vec{x}=\left( \begin{array}{c} a_{11} x_1+a_{21} x_2 \\ a_{12} x_1+a_{22} x_2 \end{array} \right) ≡ \left( \begin{array}{c} y_1 \\ y_2 \end{array} \right) \\となる\\ \\ここで\\ \frac{ \partial \vec{y} }{ \partial \vec{x} } = \left( \begin{array}{ccccc} \frac{ \partial {y}_{1} }{ \partial {x}_{1} } & \cdots & \frac{ \partial {y}_{j} }{ \partial {x}_{1} } & \cdots & \frac{ \partial {y}_{m} }{ \partial {x}_{1} }\\ \vdots & \ddots & & & \vdots \\ \frac{ \partial {y}_{1} }{ \partial {x}_{i} } & & \frac{ \partial {y}_{j} }{ \partial {x}_{i} } & & \frac{ \partial {y}_{m} }{ \partial {x}_{i} } \\ \vdots & & & \ddots & \vdots \\ \frac{ \partial {y}_{1} }{ \partial {x}_{n} } & \cdots & \frac{ \partial {y}_{j} }{ \partial {x}_{n} } & \cdots & \frac{ \partial {y}_{m} }{ \partial {x}_{n} }\end{array} \right) \\というベクトルに対するベクトル微分の定義$
http://www.r.dl.itc.u-tokyo.ac.jp/~nakagawa/SML1/math1.pdf

qiita.com

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$\frac{ \partial \vec{y} }{ \partial \vec{x} } = \left( \begin{array}{ccc} \frac{ \partial {y}_{1} }{ \partial {x}_{1} } & \frac{ \partial {y}_{2} }{ \partial {x}_{1} } \\ \frac{ \partial {y}_{1} }{ \partial {x}_{2} } & \frac{ \partial {y}_{2} }{ \partial {x}_{2} } \end{array} \right) = \left( \begin{array}{ccc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right) = {}^t A \\ \\ \\$

$\\また\\ \vec{a}_{1}= \left( \begin{array}{ccc} a_{11} \\ a_{12} \end{array} \right) , \vec{a}_{2}= \left( \begin{array}{ccc} a_{21} \\ a_{22} \end{array} \right) \\とすると\\ A=\left( \begin{array}{ccc} \vec{a}_{1} & \vec{a}_{2} \end{array} \right) \\であるので\\ \frac{ \partial f( \vec{x}) }{ \partial \vec{x} } = \left( \begin{array}{ccc} \frac{ \partial f(\vec{x}) }{ \partial {x}_{1} } \\ \frac{ \partial f(\vec{x}) }{ \partial {x}_{2} } \end{array} \right) \\より\\ \frac{ \partial \vec{y} }{ \partial A } = \left( \begin{array}{ccc} \frac{ \partial \vec{y} }{ \partial \vec{a}_{1} } & \frac{ \partial \vec{y} }{ \partial \vec{a}_{2} } \end{array} \right) = \left( \begin{array}{ccc} \left( \begin{array}{ccc} \frac{ \partial {y}_{1} }{ \partial {a}_{11} } & \frac{ \partial {y}_{2} }{ \partial {a}_{11} } \\ \frac{ \partial {y}_{1} }{ \partial {a}_{12} } & \frac{ \partial {y}_{2} }{ \partial {a}_{12} } \end{array} \right) && \left( \begin{array}{ccc} \frac{ \partial {y}_{1} }{ \partial {a}_{21} } & \frac{ \partial {y}_{2} }{ \partial {a}_{21} } \\ \frac{ \partial {y}_{1} }{ \partial {a}_{22} } & \frac{ \partial {y}_{2} }{ \partial {a}_{22} } \end{array} \right) \end{array} \right) \\ = \left( \begin{array}{ccc} \left( \begin{array}{ccc} x_1& 0 \\ 0 & x_1 \end{array} \right) & \left( \begin{array}{ccc} x_2& 0 \\ 0 & x_2 \end{array} \right) \end{array} \right) \\ = \left( \begin{array}{ccc} x_1 I & x_2 I \end{array} \right) = \left( \begin{array}{ccc} x_1 & x_2 \end{array} \right) I = \left( \begin{array}{ccc} x_1 & x_2 \end{array} \right) = {}^t \vec{x}$