1、矩阵函数求导符号说明 d/dx (y) 是一个向量，其第(i) 个元素是 dy(i)/dx d/dx (y) 是一个向量，其第(i) 个元素是 dy/dx(i) d/dx (yT) 是一个矩阵，其第(i,j) 个元素是dy(j)/dx(i) d/dx (Y) 是一个矩阵，其第(i,j) 个元素是dy(i,j)/dx d/dX (y) 是一个矩阵，其第(i,j) 个元素是dy/dx(i,j)注意 Hermitian 转置不能应用，因为复共轭不可解析，x,y 是向量，X，Y 是矩阵，x,y 是标量。在下面的表达中 A, B, C 是不依赖于 X 的矩阵，a,b 是不依赖于x 的向量，线性积 d/d
2、x (AYB) =A * d/dx (Y) * Bo d/dx (Ay) =A * d/dx (y) d/dx (xTA) =Ao d/dx (xT) =Io d/dx (xTa) = d/dx (aTx) = a d/dX (aTXb) = abTo d/dX (aTXa) = d/dX (aTXTa) = aaT d/dX (aTXTb) = baT d/dx (YZ) =Y * d/dx (Z) + d/dx (Y) * Z二次积 d/dx (Ax+b)TC(Dx+e) = ATC(Dx+e) + DTCT(Ax+b)o d/dx (xTCx) = (C+CT)x C: symmetri
3、c: d/dx (xTCx) = 2Cx d/dx (xTx) = 2xo d/dx (Ax+b)T (Dx+e) = AT (Dx+e) + DT (Ax+b) d/dx (Ax+b)T (Ax+b) = 2AT (Ax+b)o C: symmetric: d/dx (Ax+b)TC(Ax+b) = 2ATC(Ax+b) d/dX (aTXTXb) = X(abT + baT)o d/dX (aTXTXa) = 2XaaT d/dX (aTXTCXb) = CTXabT + CXbaTo d/dX (aTXTCXa) = (C + CT)XaaTo C:Symmetric d/dX (aTX
4、TCXa) = 2CXaaT d/dX (Xa+b)TC(Xa+b) = (C+CT)(Xa+b)aT三次积 d/dx (xTAxxT) = (A+AT)xxT+xTAxI逆 d/dx (Y-1) = -Y-1d/dx (Y)Y-1迹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)To d/dX (tr
5、(AX-1) =d/dX (tr(X-1A) = -X-TATX-T d/dX (tr(ATXBT) = d/dX (tr(BXTA) = ABo d/dX (tr(XAT) = d/dX (tr(ATX) =d/dX (tr(XTA) = d/dX (tr(AXT) = A d/dX (tr(AXBXT) = ATXBT + AXBo d/dX (tr(XAXT) = X(A+AT)o d/dX (tr(XTAX) = XT(A+AT)o d/dX (tr(AXTX) = (A+AT)X d/dX (tr(AXBX) = ATXTBT + BTXTAT C:symmetric d/dX (t
6、r(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行列式 d/dX (det(X) = d/dX (det(XT) = det(X)*X-To d/dX (det(AXB) = det(AXB)*X-To d/dX (ln(det(AXB) = X-T d/dX (det(Xk) = k*det(Xk)*X-
7、To d/dX (ln(det(Xk) = kX-T Real d/dX (det(XTCX) = det(XTCX)*(C+CT)X(XTCX)-1o C: Real,Symmetric d/dX (det(XTCX) = 2det(XTCX)* CX(XTCX)-1 C: Real,Symmetricc d/dX (ln(det(XTCX) = 2CX(XTCX)-1Jacobian如果y 是x 的函数，则dyT/dx 是y 关于x 的Jacobian 矩阵。其行列式|dyT/dx|是表示了dy 和dx 的超体积比值. Jacobian 行列式出现在变元积分中:Integral(f(y)d
8、y)=Integral(f(y(x) |dyT/dx| dx).Hessian 矩阵如果f 是x 的函数，则对称矩阵d2f/dx2 = d/dxT(df/dx)就是f(x)的Hessian 矩阵。 满足df/dx = 0 的x 的值，当Hessian 是正定、负定、不定时，就是相应的最小值、最大值、或者是鞍点。 d2/dx2 (aTx) = 0 d2/dx2 (Ax+b)TC(Dx+e) = ATCD + DTCTAo d2/dx2 (xTCx) = C+CT d2/dx2 (xTx) = 2Io d2/dx2 (Ax+b)T (Dx+e) = ATD + DTA d2/dx2 (Ax+b)T (Ax+b) = 2ATAo C: symmetric: d2/dx2 (Ax+b)TC(Ax+b) = 2ATCA
