矩阵求导公式

矩阵求导：

基本公式：

Y = A * X --> DY/DX = A'

Y = X * A --> DY/DX = A

Y = A' * X * B --> DY/DX = A * B'

Y = A' * X' * B --> DY/DX = B * A'

1. 矩阵Y对标量x求导：

相当于每个元素求导数后转置一下，注意M×N矩阵求导后变成N×M了

Y = [y(ij)] --> dY/dx = [dy(ji)/dx]

2. 标量y对列向量X求导：

注意与上面不同，这次括号内是求偏导，不转置，对N×1向量求导后还是N×1向量

y = f(x1,x2,..,xn) --> dy/dX = (Dy/Dx1,Dy/Dx2,..,Dy/Dxn)'

3. 行向量Y'对列向量X求导：

注意1×M向量对N×1向量求导后是N×M矩阵。

将Y的每一列对X求偏导，将各列构成一个矩阵。

重要结论：

dX'/dX = I

d(AX)'/dX = A'

4. 列向量Y对行向量X’求导：

转化为行向量Y’对列向量X的导数，然后转置。

注意M×1向量对1×N向量求导结果为M×N矩阵。

dY/dX' = (dY'/dX)'

5. 向量积对列向量X求导运算法则：

注意与标量求导有点不同。

d(UV')/dX = (dU/dX)V' + U(dV'/dX)

d(U'V)/dX = (dU'/dX)V + (dV'/dX)U'

重要结论：

d(X'A)/dX = (dX'/dX)A + (dA/dX)X' = IA + 0X' = A

d(AX)/dX' = (d(X'A')/dX)' = (A')' = A

d(X'AX)/dX = (dX'/dX)AX + (d(AX)'/dX)X = AX + A'X

6. 矩阵Y对列向量X求导：

将Y对X的每一个分量求偏导，构成一个超向量。

注意该向量的每一个元素都是一个矩阵。

7. 矩阵积对列向量求导法则：

d(uV)/dX = (du/dX)V + u(dV/dX)

d(UV)/dX = (dU/dX)V + U(dV/dX)

重要结论：

d(X'A)/dX = (dX'/dX)A + X'(dA/dX) = IA + X'0 = A

8. 标量y对矩阵X的导数：

类似标量y对列向量X的导数，

把y对每个X的元素求偏导，不用转置。

dy/dX = [ Dy/Dx(ij) ]

重要结论：

y = U'XV = ΣΣu(i)x(ij)v(j) 于是 dy/dX = [u(i)v(j)] = UV'

y = U'X'XU 则 dy/dX = 2XUU'

y = (XU-V)'(XU-V) 则 dy/dX = d(U'X'XU - 2V'XU + V'V)/dX = 2XUU' - 2VU' + 0 = 2(XU-V)U'

9. 矩阵Y对矩阵X的导数：

将Y的每个元素对X求导，然后排在一起形成超级矩阵。



10.乘积的导数

d(f*g)/dx=(df'/dx)g+(dg/dx)f'

结论

d(x'Ax)=(d(x'')/dx)Ax+(d(Ax)/dx)(x'')=Ax+A'x （注意：''是表示两次转置）

比较详细点的如下：




















http://lzh21cen.blog.163.com/blog/static/145880136201051113615571/ http://hi.baidu.com/wangwen926/blog/item/eb189bf6b0fb702b720eec94.html
其他参考：

Contents

Notation
Derivatives of Linear Products
Derivatives of Quadratic Products

Notation

d/dx (y) is a vector whose (i) element is dy(i)/dx
d/dx (y) is a vector whose (i) element is dy/dx(i)
d/dx (yT) is a matrix whose (i,j) element is dy(j)/dx(i)
d/dx (Y) is a matrix whose (i,j) element is dy(i,j)/dx
d/dX (y) is a matrix whose (i,j) element is dy/dx(i,j)

Note that the Hermitian transpose is not used because complex conjugates are not analytic.

In the expressions below matrices and vectors A, B, C do not depend on X.

Derivatives of Linear Products

d/dx (AYB) =A * d/dx (Y) * B

d/dx (Ay) =A * d/dx (y)

d/dx (xTA) =A

d/dx (xT) =I
d/dx (xTa) = d/dx (aTx)
= a

d/dX (aTXb) = abT

d/dX (aTXa) = d/dX (aTXTa)
= aaT

d/dX (aTXTb)
= baT
d/dx (YZ) =Y * d/dx (Z) + d/dx (Y) *
Z

Derivatives of Quadratic Products

d/dx (Ax+b)TC(Dx+e)
= ATC(Dx+e) + DTCT(Ax+b)

d/dx (xTCx) = (C+CT)x

[C: symmetric]: d/dx (xTCx)
= 2Cx
d/dx (xTx) = 2x

d/dx (Ax+b)T (Dx+e)
= AT (Dx+e) + DT (Ax+b)

d/dx (Ax+b)T (Ax+b)
= 2AT (Ax+b)

[C: symmetric]: d/dx (Ax+b)TC(Ax+b)
= 2ATC(Ax+b)

d/dX (aTXTXb)
= X(abT + baT)

d/dX (aTXTXa)
= 2XaaT

d/dX (aTXTCXb)
= CTXabT + CXbaT

d/dX (aTXTCXa)
= (C + CT)XaaT
[C:Symmetric] d/dX (aTXTCXa)
= 2CXaaT

d/dX ((Xa+b)TC(Xa+b)) = (C+CT)(Xa+b)aT

Derivatives of Cubic Products

d/dx (xTAxxT) = (A+AT)xxT+xTAxI

Derivatives of Inverses

d/dx (Y-1) = -Y-1d/dx (Y)Y-1

Derivative of Trace

Note: matrix dimensions must result in an n*n argument for tr().

d/dX (tr(X)) = I
d/dX (tr(Xk)) =k(Xk-1)T
d/dX (tr(AXk)) = SUMr=0:k-1(XrAXk-r-1)T
d/dX (tr(AX-1B)) = -(X-1BAX-1)T

d/dX (tr(AX-1)) =d/dX (tr(X-1A))
= -X-TATX-T

d/dX (tr(ATXBT))
= d/dX (tr(BXTA)) = AB

d/dX (tr(XAT)) = d/dX (tr(ATX))
=d/dX (tr(XTA)) = d/dX (tr(AXT)) = A

d/dX (tr(AXBXT)) = ATXBT + AXB

d/dX (tr(XAXT)) = X(A+AT)
d/dX (tr(XTAX)) = XT(A+AT)
d/dX (tr(AXTX)) = (A+AT)X

d/dX (tr(AXBX)) = ATXTBT + BTXTAT
[C:symmetric] d/dX (tr((XTCX)-1A)
= d/dX (tr(A (XTCX)-1) = -(CX(XTCX)-1)(A+AT)(XTCX)-1
[B,C:symmetric] d/dX (tr((XTCX)-1(XTBX))
= d/dX (tr( (XTBX)(XTCX)-1) = -2(CX(XTCX)-1)XTBX(XTCX)-1 +
2BX(XTCX)-1

Derivative of Determinant

Note: matrix dimensions must result in an n*n argument for det().

d/dX (det(X)) = d/dX (det(XT))
= det(X)*X-T

d/dX (det(AXB)) = det(AXB)*X-T
d/dX (ln(det(AXB))) = X-T

d/dX (det(Xk)) = k*det(Xk)*X-T

d/dX (ln(det(Xk))) = kX-T

[Real] d/dX (det(XTCX))
= det(XTCX)*(C+CT)X(XTCX)-1

[C: Real,Symmetric] d/dX (det(XTCX))
= 2det(XTCX)* CX(XTCX)-1

[C: Real,Symmetricc] d/dX (ln(det(XTCX)))
= 2CX(XTCX)-1

Jacobian

If y is a function of x, then dyT/dx is the Jacobian matrix of y with
respect to x.

Its determinant, |dyT/dx|, is the Jacobian of y with respect to x and represents
the ratio of the hyper-volumes dy and dx. The Jacobian occurs when changing variables in an integration: Integral(f(y)dy)=Integral(f(y(x))
|dyT/dx| dx).

Hessian matrix

If f is a function of x then the symmetric matrix d2f/dx2 = d/dxT(df/dx)
is the Hessian matrix of f(x). A value of x for which df/dx = 0 corresponds to a minimum, maximum or saddle point according to whether the Hessian
is positive definite, negative definite or indefinite.

d2/dx2 (aTx) = 0
d2/dx2 (Ax+b)TC(Dx+e)
= ATCD + DTCTA

d2/dx2 (xTCx) = C+CT

d2/dx2 (xTx) = 2I

d2/dx2 (Ax+b)T (Dx+e)
= ATD + DTA

d2/dx2 (Ax+b)T (Ax+b)
= 2ATA

[C: symmetric]: d2/dx2 (Ax+b)TC(Ax+b)
= 2ATCA
http://www.psi.toronto.edu/matrix/calculus.html
http://www.stanford.edu/~dattorro/matrixcalc.pdf

http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.AppD.d/IFEM.AppD.pdf

http://www4.ncsu.edu/~pfackler/MatCalc.pdf

http://center.uvt.nl/staff/magnus/wip12.pdf