daruma3940の日記

理解や文章に間違い等あればどんなことでもご指摘お願いします

行列 ベクトル 微分

f:id:daruma3940:20160520223745p:plain
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

から

\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}