原文：http://blog.sina.com.cn/s/blog_7959e7ed0100w2b3.html
今天推導公式，發現居然有對矩陣的求導，狂汗–完全不會。不過還好網上有人總結了。吼吼，趕緊搬過來收藏備份。
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’
- 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]
- 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)’
- 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’
- 列向量Y對行向量X’求導：
轉化爲行向量Y’對列向量X的導數，然後轉置。
注意M×1向量對1×N向量求導結果爲M×N矩陣。
dY/dX’ = (dY’/dX)’
- 向量積對列向量X求導運算法則：
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
- 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。
- 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
- 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’
- 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）
其他參考：
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)) = kdet(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