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. 矩陣Y對標量x求導：

相當於每個元素求導數後轉置一下，注意M×N矩陣求導後變成N×M了

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

2. 標量y對列向量X求導：

注意與上面不同，這次括號內是求偏導，不轉置，對N×1向量求導後還是N×1向量

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

3. 行向量Y'對列向量X求導：

注意1×M向量對N×1向量求導後是N×M矩陣。

將Y的每一列對X求偏導，將各列構成一個矩陣。

重要結論：

dX'/dX = I

d(AX)'/dX = A'

4. 列向量Y對行向量X’求導：

轉化為行向量Y’對列向量X的導數，然後轉置。

注意M×1向量對1×N向量求導結果為M×N矩陣。

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

5. 向量積對列向量X求導運演算法則：

注意與標量求導有點不同。

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

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

重要結論：

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. 矩陣Y對列向量X求導：

將Y對X的每一個分量求偏導，構成一個超向量。

注意該向量的每一個元素都是一個矩陣。

7. 矩陣積對列向量求導法則：

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

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

重要結論：

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

8. 標量y對矩陣X的導數：

類似標量y對列向量X的導數，

把y對每個X的元素求偏導，不用轉置。

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

重要結論：

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

y = U'X'XU 則 dy/dX = 2XUU'

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

9. 矩陣Y對矩陣X的導數：

將Y的每個元素對X求導，然後排在一起形成超級矩陣。

10.乘積的導數

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

結論

d(x'Ax)=(d(x'')/dx)Ax+(d(Ax)/dx)(x'')=Ax+A'x  （注意：''是表示兩次轉置）

比較詳細點的如下：

http://lzh21cen.blog.163.com/blog/static/145880136201051113615571/

http://hi.baidu.com/wangwen926/blog/item/eb189bf6b0fb702b720eec94.html

其他參考：

Contents

• Notation
• Derivatives of Linear Products
• Derivatives of Quadratic Products

Notation

• d/dx (y)isa vector whose (i) elementis dy(i)/dx
• d/dx (y) is a vectorwhose (i) elementis dy/dx(i)
• d/dx (y^T)is a matrixwhose (i,j) elementis dy(j)/dx(i)
• d/dx (Y) is a matrixwhose (i,j) elementis dy(i,j)/dx
• d/dX (y) is a matrixwhose (i,j) elementis dy/dx(i,j)

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

In the expressions below matrices andvectors A, B, C donot depend on X.

Derivatives of Linear Products

• d/dx (AYB) =A * d/dx (Y)* B
• • d/dx (Ay) =A * d/dx (y)
• d/dx(x^TA) =A
• • d/dx(x^T) =I
  • d/dx(x^Ta) =d/dx(a^Tx)= a
• d/dX(a^TXb)= ab^T
• • d/dX(a^TXa)= d/dX(a^TX^Ta)= aa^T
• d/dX(a^TX^Tb)= ba^T
• d/dx (YZ) =Y * d/dx (Z)+ d/dx (Y) *Z

Derivatives of Quadratic Products

• d/dx (Ax+b)^TC(Dx+e)= A^TC(Dx+e) + D^TC^T(Ax+b)
• • d/dx (x^TCx)= (C+C^T)x
  • • [C:symmetric]: d/dx (x^TCx)= 2Cx
    • d/dx (x^Tx)= 2x
  • d/dx (Ax+b)^T (Dx+e)= A^T (Dx+e) + D^T (Ax+b)
  • • d/dx (Ax+b)^T (Ax+b)= 2A^T (Ax+b)
  • [C:symmetric]: d/dx (Ax+b)^TC(Ax+b)= 2A^TC(Ax+b)
• d/dX(a^TX^TXb)= X(ab^T +ba^T)
• • d/dX(a^TX^TXa)= 2Xaa^T
• d/dX(a^TX^TCXb)= C^TXab^T +CXba^T
• • d/dX(a^TX^TCXa)= (C +C^T)Xaa^T
  • [C:Symmetric] d/dX(a^TX^TCXa)= 2CXaa^T
• d/dX((Xa+b)^TC(Xa+b))=(C+C^T)(Xa+b)a^T

Derivatives of Cubic Products

• d/dx(x^TAxx^T)=(A+A^T)xx^T+x^TAxI

Derivatives of Inverses

• d/dx (Y^-1)= -Y^-1d/dx (Y)Y^-1

Derivative of Trace

Note: matrix dimensions must result inan n*n argument fortr().

• d/dX(tr(X))= I
• d/dX(tr(X^k))=k(X^k-1)^T
• d/dX(tr(AX^k))=SUM_r=0:k-1(X^rAX^k-r-1)^T
• d/dX(tr(AX^-1B))= -(X^-1BAX^-1)^T
  • d/dX(tr(AX^-1))=d/dX(tr(X^-1A))= -X^-TA^TX^-T
• d/dX(tr(A^TXB^T))= d/dX(tr(BX^TA))= AB
  • d/dX(tr(XA^T))= d/dX(tr(A^TX))=d/dX(tr(X^TA))= d/dX(tr(AX^T)) =A
• d/dX(tr(AXBX^T))= A^TXB^T + AXB
  • d/dX(tr(XAX^T))= X(A+A^T)
  • d/dX(tr(X^TAX))= X^T(A+A^T)
  • d/dX(tr(AX^TX))= (A+A^T)X
• d/dX(tr(AXBX))= A^TX^TB^T + B^TX^TA^T
• [C:symmetric] d/dX(tr((X^TCX)^-1A)= d/dX(tr(A(X^TCX)^-1)= -(CX(X^TCX)^-1)(A+A^T)(X^TCX)^-1
• [B,C:symmetric] d/dX(tr((X^TCX)^-1(X^TBX))= d/dX(tr((X^TBX)(X^TCX)^-1)= -2(CX(X^TCX)^-1)X^TBX(X^TCX)^-1 +2BX(X^TCX)^-1

Derivative of Determinant

Note: matrix dimensions must result inan n*n argument fordet().

• d/dX(det(X))= d/dX(det(X^T))= det(X)*X^-T
• • d/dX(det(AXB)) =det(AXB)*X^-T
  • d/dX(ln(det(AXB)))= X^-T
• d/dX(det(X^k))= k*det(X^k)*X^-T
  • d/dX(ln(det(X^k)))= kX^-T
• [Real] d/dX(det(X^TCX))=det(X^TCX)*(C+C^T)X(X^TCX)^-1
• • [C: Real,Symmetric] d/dX(det(X^TCX))= 2det(X^TCX)*CX(X^TCX)^-1
• [C: Real,Symmetricc] d/dX(ln(det(X^TCX)))= 2CX(X^TCX)^-1

If y is a functionof x,then dy^T/dx isthe Jacobian matrixof y with respectto x.

Its determinant,|dy^T/dx|, isthe Jacobian of y withrespect to x andrepresents the ratio of thehyper-volumes dy and dx.The Jacobian occurs when changing variables in an integration:Integral(f(y)dy)=Integral(f(y(x))|dy^T/dx|dx).

Hessian matrix

If f is a functionof x then the symmetricmatrixd²f/dx² = d/dx^T(df/dx)is the Hessian matrix off(x). A valueof x for whichdf/dx = 0 correspondsto a minimum, maximum or saddle point according to whether theHessian is positive definite, negative definite or indefinite.

• d²/dx² (a^Tx)= 0
• d²/dx² (Ax+b)^TC(Dx+e)= A^TCD + D^TC^TA
• • d²/dx² (x^TCx)= C+C^T
  • • d²/dx² (x^Tx)= 2I
  • d²/dx² (Ax+b)^T (Dx+e)= A^TD + D^TA
  • • d²/dx² (Ax+b)^T (Ax+b)= 2A^TA
  • [C:symmetric]: d²/dx² (Ax+b)^TC(Ax+b)= 2A^TCA