矩阵求导公式..docx

转载]矩阵求导公式【转】? (2011-11-15 11:03:34)javascript:;转载▼标签：?/?c=blogq=%D7%AA%D4%D8by=tag转载原文地址：/s/blog_6c17a3a00100qg5w.html矩阵求导公式【转】作者：/u/1813488544三寅今天推导公式，发现居然有对矩阵的求导，狂汗--完全不会。不过还好网上有人总结了。吼吼，赶紧搬过来收藏备份。基本公式：Y = A * X -- DY/DX = AY = X * A -- DY/DX = AY = A * X * B -- DY/DX = A * BY = A * X * B -- DY/DX = B * A1. 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 = Id(AX)/dX = A4. 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(UV)/dX = (dU/dX)V + (dV/dX)U重要结论：d(XA)/dX = (dX/dX)A + (dA/dX)X = IA + 0X = Ad(AX)/dX = (d(XA)/dX) = (A) = Ad(XAX)/dX = (dX/dX)AX + (d(AX)/dX)X = AX + AX6. 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(XA)/dX = (dX/dX)A + X(dA/dX) = IA + X0 = A8. 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 = UXV = ΣΣu(i)x(ij)v(j) 于是 dy/dX = [u(i)v(j)] = UVy = UXXU 则 dy/dX = 2XUUy = (XU-V)(XU-V) 则 dy/dX = d(UXXU - 2VXU + VV)/dX = 2XUU - 2VU + 0 = 2(XU-V)U9. 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(xAx)=(d(x)/dx)Ax+(d(Ax)/dx)(x)=Ax+Ax（注意：是表示两次转置）比较详细点的如下：?/blog/static/145880136201051113615571//wangwen926/blog/item/eb189bf6b0fb702b720eec94.html其他参考：?ContentsNotationDerivatives of Linear ProductsDerivatives of Quadratic ProductsNotationd/dx?(y)?is a vector whose?(i)?element is?dy(i)/dxd/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)/dxd/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?