函数矩阵对矩阵求导
函数对矩阵以及函数矩阵对矩阵求导，我理解主要就是一种简化的写法，用矩阵将多个多元函数对每个元求导写成矩阵的形式，看起来比较简洁。函数对矩阵的导数
设矩阵 X = ( ξ i j ) m × n \mathbf{X}=({\xi_{ij}})_{m\times n} X=(ξij)m×n， m n mn mn元函数 f ( X ) = f ( ξ 11 , ξ 12 , ξ 13 , … , ξ m 1 , … , ξ m n ) f(\mathbf{X})=f(\xi_{11},\xi_{12},\xi_{13},\dots,\xi_{m1},\dots,\xi_{mn}) f(X)=f(ξ11,ξ12,ξ13,…,ξm1,…,ξmn)，则 f ( X ) f(\mathbf{X}) f(X)对矩阵 X \mathbf{X} X的导数为，
d f d X = ( ∂ f ∂ ξ i j ) m × n = [ ∂ f ∂ ξ 11 ∂ f ∂ ξ 12 … ∂ f ∂ ξ 1 n ⋮ ⋮ ⋮ ∂ f ∂ ξ m 1 ∂ f ∂ ξ m 2 … ∂ f ∂ ξ m n ] \dfrac {df}{d\mathbf{X}}=\left( \dfrac {\partial f}{\partial \xi _{ij}}\right) _{m\times n}=\begin{bmatrix} \dfrac {\partial f}{\partial \xi_{11}} & \dfrac {\partial f}{\partial \xi_{12}} & \ldots & \dfrac {\partial f}{\partial \xi_{1n}} \\ \vdots & \vdots & & \vdots \\ \dfrac {\partial f}{\partial \xi_{m1}} & \dfrac {\partial f}{\partial \xi_{m2}} & \ldots & \dfrac {\partial f}{\partial \xi_{mn}} \end{bmatrix} dXdf=(∂ξij∂f)m×n=⎣⎢⎢⎢⎢⎡∂ξ11∂f⋮∂ξm1∂f∂ξ12∂f⋮∂ξm2∂f……∂ξ1n∂f⋮∂ξmn∂f⎦⎥⎥⎥⎥⎤
例如， X = ( ξ 1 , ξ 2 , … , ξ n ) T \mathbf{X}=(\xi_1,\xi_2,\dots,\xi_n)^{T} X=(ξ1,ξ2,…,ξn)T， n n n元函数 f ( X ) = f ( ξ 1 , ξ 2 , … , ξ n ) f(\mathbf{X})=f(\xi_1,\xi_2,\dots,\xi_n) f(X)=f(ξ1,ξ2,…,ξn)，
则，
d f d X = ( d f d ξ 1 , d f d ξ 2 , … , d f ξ n ) T \dfrac{df}{d\mathbf{X}}=(\dfrac{df}{d\xi_1},\dfrac{df}{d\xi_2},\dots,\dfrac{df}{\xi_n})^{T} dXdf=(dξ1df,dξ2df,…,ξndf)T
又有，
d f d X T = ( d f d ξ 1 , d f d ξ 2 , … , d f ξ n ) \dfrac{df}{d{\mathbf{X}}^{T}}=(\dfrac{df}{d\xi_1},\dfrac{df}{d\xi_2},\dots,\dfrac{df}{\xi_n}) dXTdf=(dξ1df,dξ2df,…,ξndf)
函数矩阵对矩阵的导数
设矩阵 X = ( ξ i j ) m × n \mathbf{X}=({\xi_{ij}})_{m\times n} X=(ξij)m×n， m n mn mn元函数 f i j ( X ) = f i j ( ξ 11 , ξ 12 , ξ 13 , … , ξ m 1 , … , ξ m n ) ( i = 1 , 2 , 3 … , r ; j = 1 , 2 , … , s ) f_{ij}(\mathbf{X})=f_{ij}(\xi_{11},\xi_{12},\xi_{13},\dots,\xi_{m1},\dots,\xi_{mn})(i=1,2,3\dots,r;j=1,2,\dots,s) fij(X)=fij(ξ11,ξ12,ξ13,…,ξm1,…,ξmn)(i=1,2,3…,r;j=1,2,…,s),定义函数矩阵，
F ( X ) = [ f 11 ( X ) … f 1 s ( X ) ⋮ ⋮ ⋮ f r 1 ( X ) … f r s ( X ) ] \mathbf{F}(\mathbf{X})=\begin{bmatrix} {f_{11}(\mathbf{X} )} & \ldots & {f_{1s}(\mathbf{X}}) \\ \vdots & \vdots & \vdots \\ {f_{r1}(\mathbf{X} )} & \dots & {f_{rs}(\mathbf{X})} \end{bmatrix} F(X)=⎣⎢⎡f11(X)⋮fr1(X)…⋮…f1s(X)⋮frs(X)⎦⎥⎤
对矩阵 X \mathbf{X} X的导数为，
d F d X = [ ∂ F ∂ ξ 11 ∂ F ∂ ξ 12 … ∂ F ∂ ξ 1 n ⋮ ⋮ ⋮ ∂ F ∂ ξ m 1 ∂ F ∂ ξ m 2 … ∂ F ∂ ξ m n ] \dfrac {d\mathbf{F}}{d\mathbf{X}}=\begin{bmatrix} \dfrac {\partial \mathbf{F}}{\partial \xi_{11}} & \dfrac {\partial \mathbf{F}}{\partial \xi_{12}} & \ldots & \dfrac {\partial \mathbf{F}}{\partial \xi_{1n}} \\ \vdots & \vdots & & \vdots \\ \dfrac {\partial \mathbf{F}}{\partial \xi_{m1}} & \dfrac {\partial \mathbf{F}}{\partial \xi_{m2}} & \ldots & \dfrac {\partial \mathbf{F}}{\partial \xi_{mn}} \end{bmatrix} dXdF=⎣⎢⎢⎢⎢⎡∂ξ11∂F⋮∂ξm1∂F∂ξ12∂F⋮∂ξm2∂F……∂ξ1n∂F⋮∂ξmn∂F⎦⎥⎥⎥⎥⎤
其中，
∂ F ∂ ξ i j = [ ∂ f 11 ∂ ξ i j ∂ f 12 ∂ ξ i j … ∂ f 1 s ∂ ξ i j ⋮ ⋮ ⋮ ∂ f r 1 ∂ ξ i j ∂ f r 1 ∂ ξ i j … ∂ f r s ∂ ξ i j ] \dfrac {\partial \mathbf{F}}{\partial \xi _{ij}}=\begin{bmatrix} \dfrac {\partial f_{11}}{\partial \xi_{ij}} & \dfrac {\partial f_{12}}{\partial \xi_{ij}} & \ldots & \dfrac {\partial f_{1s}}{\partial \xi _{ij}} \\ \vdots & \vdots & & \vdots \\ \dfrac {\partial f_{r1}}{\partial \xi_{ij}} & \dfrac {\partial f_{r1}}{\partial \xi_{ij}} & \ldots & \dfrac {\partial f_{rs}}{\partial \xi_{ij}} \end{bmatrix} ∂ξij∂F=⎣⎢⎢⎢⎢⎢⎡∂ξij∂f11⋮∂ξij∂fr1∂ξij∂f12⋮∂ξij∂fr1……∂ξij∂f1s⋮∂ξij∂frs⎦⎥⎥⎥⎥⎥⎤
例如， X = ( ξ 1 , ξ 2 , … , ξ n ) T \mathbf{X}=(\xi_1,\xi_2,\dots,\xi_n)^{T} X=(ξ1,ξ2,…,ξn)T， n n n元函数 f ( X ) = f ( ξ 1 , ξ 2 , … , ξ n ) f(\mathbf{X})=f(\xi_1,\xi_2,\dots,\xi_n) f(X)=f(ξ1,ξ2,…,ξn)，
则，
d f d X = ( d f d ξ 1 , d f d ξ 2 , … , d f ξ n ) T \dfrac{df}{d\mathbf{X}}=(\dfrac{df}{d\xi_1},\dfrac{df}{d\xi_2},\dots,\dfrac{df}{\xi_n})^{T} dXdf=(dξ1df,dξ2df,…,ξndf)T
因此，
d d X T ( d f d X ) = [ ∂ 2 f ∂ ξ 1 2 ∂ 2 f ∂ ξ 1 ∂ ξ 2 ⋯ ∂ 2 f ∂ ξ 1 ∂ ξ n ∂ 2 f ∂ ξ 2 ∂ ξ 1 ∂ 2 f ∂ ξ 2 2 ⋯ ∂ 2 f ∂ ξ 2 ∂ ξ n ⋮ ⋮ ⋮ ⋮ ∂ 2 f ∂ ξ n ∂ ξ 1 ∂ 2 f ∂ ξ n ∂ ξ 2 ⋯ ∂ 2 f ∂ ξ n 2 ] \dfrac {d}{d\mathbf{X}^{T}}\left( \dfrac {df}{d\mathbf{X}}\right) =\begin{bmatrix}\dfrac {\partial ^{2}f}{\partial \xi_{1}^2} & \dfrac {\partial^{2} f}{\partial \xi _{1}\partial \xi _{2}} & \cdots & \dfrac {\partial^{2} f}{\partial \xi _{1}\partial \xi _{n}}\\ \dfrac {\partial ^{2}f}{\partial \xi_{2}\partial \xi_{1}} & \dfrac {\partial^{2} f}{{\partial \xi _{2}}^{2}} & \cdots & \dfrac {\partial^{2} f}{\partial \xi _{2}\partial \xi _{n}}\\ \vdots &\vdots & \vdots& \vdots\\ \dfrac {\partial ^{2}f}{\partial \xi_{n}\partial \xi_{1}} & \dfrac {\partial^{2} f}{\partial \xi_{n}\partial \xi _{2}} & \cdots & \dfrac {\partial^{2} f}{\partial {\xi _{n}}^{2}} \end{bmatrix} dXTd(dXdf)=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡∂ξ12∂2f∂ξ2∂ξ1∂2f⋮∂ξn∂ξ1∂2f∂ξ1∂ξ2∂2f∂ξ22∂2f⋮∂ξn∂ξ2∂2f⋯⋯⋮⋯∂ξ1∂ξn∂2f∂ξ2∂ξn∂2f⋮∂ξn2∂2f⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤
页:
[1]