矩阵对标量求导：
U=U(x,y)U = U(x,y)U=U(x,y)
U 是矩阵, x,y 是标量.
∂∣U∣∂x=∣U∣tr(U−1∂U∂x)\frac{\partial |U|}{\partial x}=|U|tr(U^{-1}{\frac{\partial U}{\partial x}})∂x∂∣U∣=∣U∣tr(U−1∂x∂U)
∂U−1∂x=−U−1∂U∂xU−1\frac{\partial U^{-1}}{\partial x}=-U^{-1}{\frac{\partial U}{\partial x}}U^{-1}∂x∂U−1=−U−1∂x∂UU−1
∂ln∣U∣∂x=tr(U−1∂U∂x)\frac{\partial ln|U|}{\partial x}=tr(U^{-1}{\frac{\partial U}{\partial x}})∂x∂ln∣U∣=tr(U−1∂x∂U)
∂2∣U∣∂x2=∣U∣(tr(U−1∂2U∂x2)+(tr(U−1∂U∂x))2−tr(U−1∂U∂xU−1∂U∂x)) \frac{\partial^2 |U|}{\partial x^2}=|U|(tr(U^{-1}{\frac{\partial^2 U}{\partial x^2}})+(tr(U^{-1}{\frac{\partial U}{\partial x}}))^2 - tr(U^{-1}{\frac{\partial U}{\partial x}}U^{-1}{\frac{\partial U}{\partial x}}))∂x2∂2∣U∣=∣U∣(tr(U−1∂x2∂2U)+(tr(U−1∂x∂U))2−tr(U−1∂x∂UU−1∂x∂U))
∂2U∂x2=U−1(∂U∂xU−1∂U∂x−∂2U∂x2+∂U∂xU−1∂U∂x)U−1\frac{\partial^2 U}{\partial x^2}= U^{-1}({\frac{\partial U}{\partial x}}U^{-1}{\frac{\partial U}{\partial x}} - {\frac{\partial^2 U}{\partial x^2}} + {\frac{\partial U}{\partial x}}U^{-1}{\frac{\partial U}{\partial x}})U^{-1}∂x2∂2U=U−1(∂x∂UU−1∂x∂U−∂x2∂2U+∂x∂UU−1∂x∂U)U−1
∂2U∂x∂y=U−1(∂U∂xU−1∂U∂y−∂2U∂x∂y+∂U∂yU−1∂U∂x)U−1\frac{\partial^2 U}{\partial x\partial y}= U^{-1}({\frac{\partial U}{\partial x}}U^{-1}{\frac{\partial U}{\partial y}} - {\frac{\partial^2 U}{\partial x \partial y}} + {\frac{\partial U}{\partial y}}U^{-1}{\frac{\partial U}{\partial x}})U^{-1}∂x∂y∂2U=U−1(∂x∂UU−1∂y∂U−∂x∂y∂2U+∂y∂UU−1∂x∂U)U−1
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