在对向量进行导数运算时非常有效的三个公式~
在进行对向量的求导时,非常好用的三个公式
分别是
1.对于向量x求导
∇
x
w
T
x
=
w
\nabla_x w^Tx=w
∇xwTx=w
2.对向量x求导
∇
x
x
T
A
x
=
(
A
+
A
T
)
x
\nabla_x x^TAx=(A+A^T)x
∇xxTAx=(A+AT)x其中x为向量,A为矩阵
3.对向量x求二阶导(即Hessian矩阵)
∇
2
x
T
A
x
=
A
+
A
T
\nabla ^2 x^TAx=A+A^T
∇2xTAx=A+AT
详细的证明
1.对于向量x求导
∇
x
w
T
x
=
w
\nabla_x w^Tx=w
∇xwTx=w
证明:
w
T
x
=
(
w
1
w
2
.
.
.
w
n
)
⋅
(
x
1
x
2
.
.
.
x
n
)
=
∑
i
=
1
n
w
i
x
i
w^Tx=\begin{pmatrix}w_1&w_2&...&w_n\end{pmatrix}\cdot\begin{pmatrix}x_1\\x_2\\...\\x_n\end{pmatrix}\\ =\sum\limits_{i=1}^nw_ix_i
wTx=(w1w2...wn)⋅⎝⎜⎜⎛x1x2...xn⎠⎟⎟⎞=i=1∑nwixi
所以对
x
i
x_i
xi求导,对应的导数为
w
i
w_i
wi
故
∇
x
w
T
x
=
w
\nabla_x w^Tx=w
∇xwTx=w
2.对向量x求导
∇
x
x
T
A
x
=
(
A
+
A
T
)
x
\nabla_x x^TAx=(A+A^T)x
∇xxTAx=(A+AT)x
其中x为向量,A为矩阵
证明:
对于二次型
x
T
A
x
x^TAx
xTAx
x
T
A
x
=
(
x
1
x
2
.
.
.
x
n
)
(
a
11
a
12
.
.
.
a
1
n
a
21
a
22
.
.
.
a
2
n
.
.
a
n
1
a
n
2
.
.
.
a
n
n
)
(
x
1
x
2
.
.
.
x
n
)
=
(
x
1
x
2
.
.
.
x
n
)
(
a
11
x
1
+
a
12
x
2
+
.
.
.
+
a
1
n
x
n
a
21
x
1
+
a
22
x
2
+
.
.
.
+
a
2
n
x
n
.
.
.
a
n
1
x
1
+
a
n
2
x
2
+
.
.
.
+
a
n
n
x
n
)
=
a
11
x
1
x
1
+
a
12
x
1
x
2
+
.
.
.
+
a
1
n
x
1
x
n
+
a
21
x
2
x
1
+
a
22
x
2
x
2
+
.
.
.
+
a
2
n
x
2
x
n
+
.
.
.
+
a
n
1
x
n
x
1
+
a
n
2
x
n
x
2
+
.
.
.
+
a
n
n
x
n
x
n
=
∑
i
=
1
n
∑
j
=
1
n
a
i
j
x
i
x
j
x^TAx=\begin{pmatrix}x_1&x_2&...&x_n\end{pmatrix}\begin{pmatrix} a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\.\\.\\a_{n1}&a_{n2}&...&a_{nn}\end{pmatrix}\begin{pmatrix}x_1\\x_2\\...\\\\x_n\end{pmatrix}\\ =\begin{pmatrix}x_1&x_2&...&x_n\end{pmatrix}\begin{pmatrix}a_{11}x_1+a_{12}x_2+...+a_{1n}x_{n}\\a_{21}x_1+a_{22}x_2+...+a_{2n}x_n\\...\\a_{n1}x_1+a_{n2}x_2+...+a_{nn}x_n\end{pmatrix}\\ =a_{11}x_1x_1+a_{12}x_1x_2+...+a_{1n}x_1x_n+a_{21}x_2x_1+a_{22}x_2x_2+...+a_{2n}x_2x_n +...+a_{n1}x_nx_1+a_{n2}x_nx_2+...+a_{nn}x_nx_n \\ =\sum\limits_{i=1}^n\sum\limits_{j=1}^na_{ij}x_ix_j
xTAx=(x1x2...xn)⎝⎜⎜⎜⎜⎛a11a21..an1a12a22an2.........a1na2nann