y = x T A x y=x^TAx y=xTAx,其中x是n维向量，A是n阶方阵，求 d y / d x dy/dx dy/dx 记 A = [ a i j ] A=\left[a_{i j}\right] A=[aij​]. x ∈ R n , x = ( x 1 , … , x n ) T x \in \mathbb{R}^{n}, x=\left(x_{1}, \ldots, x_{n}\right)^{T} x∈Rn,x=(x1​,…,xn​)T, 则 y = ∑ i = 1 n ∑ j = 1 n a i j x i x j y=\sum_{i=1}^{n} \sum_{j=1}^{n} a_{i j} x_{i} x_{j} y=∑i=1n​∑j=1n​aij​xi​xj​ 故 ∂ y ∂ x k = ∑ i ≠ k ∂ ∂ x k ( ∑ j = 1 n a i j x i x j ) + ∂ ∂ x k ( ∑ j = 1 n a k j x k x j ) = ∑ i ≠ k ( ∂ ∂ x k ( ∑ j ≠ k a i j x i x j ) + ∂ ∂ x k ( a i k x i x k ) ) + ∑ j ≠ k ∂ ∂ x k ( a k j x k x j ) + ∂ ∂ x k ( a k k x k 2 ) = ∑ i ≠ k ( 0 + a i k x i ) + ∑ j ≠ k a k j x j + 2 a k k x k = ∑ i = 1 n a i k x i + ∑ j = 1 n a k j x j = ( x T A ) k + ( A x ) k \begin{aligned} \frac{\partial y}{\partial x_{k}} &=\sum_{i \neq k} \frac{\partial}{\partial x_{k}}\left(\sum_{j=1}^{n} a_{i j} x_{i} x_{j}\right)+\frac{\partial}{\partial x_{k}}\left(\sum_{j=1}^{n} a_{k j} x_{k} x_{j}\right) \\ &=\sum_{i \neq k}\left(\frac{\partial}{\partial x_{k}}\left(\sum_{j \neq k} a_{i j} x_{i} x_{j}\right)+\frac{\partial}{\partial x_{k}}\left(a_{i k} x_{i} x_{k}\right)\right)+\sum_{j \neq k} \frac{\partial}{\partial x_{k}}\left(a_{k j} x_{k} x_{j}\right)+\frac{\partial}{\partial x_{k}}\left(a_{k k} x_{k}^{2}\right) \\ &=\sum_{i \neq k}( 0+a_{i k} x_{i})+\sum_{j \neq k} a_{k j} x_{j}+2 a_{k k} x_{k} \\ &=\sum_{i=1}^{n} a_{i k} x_{i}+\sum_{j=1}^{n} a_{k j} x_{j} \\ &=\left(x^{T} A\right)_{k}+(A x)_{k} \end{aligned} ∂xk​∂y​​=i​=k∑​∂xk​∂​(j=1∑n​aij​xi​xj​)+∂xk​∂​(j=1∑n​akj​xk​xj​)=i​=k∑​⎝⎛​∂xk​∂​⎝⎛​j​=k∑​aij​xi​xj​⎠⎞​+∂xk​∂​(aik​xi​xk​)⎠⎞​+j​=k∑​∂xk​∂​(akj​xk​xj​)+∂xk​∂​(akk​xk2​)=i​=k∑​(0+aik​xi​)+j​=k∑​akj​xj​+2akk​xk​=i=1∑n​aik​xi​+j=1∑n​akj​xj​=(xTA)k​+(Ax)k​​ 其中 ( x T A ) k \left(x^{T} A\right)_{k} (xTA)k​ 是行向量 x T A x^{T} A xTA的第k个分量， ( A x ) k (A x)_{k} (Ax)k​是列向量 A x Ax Ax的第k个分量。因此 ∂ y ∂ x k = ( x T A ) k + ( x T A T ) k \frac{\partial y}{\partial x_{k}}=\left(x^{T} A\right)_{k}+\left(x^{T} A^{T}\right)_{k} ∂xk​∂y​=(xTA)k​+(xTAT)k​. 所以 ∇ y = x T A + x T A T = x T ( A + A T ) \nabla y=x^{T} A+x^{T} A^{T}=x^{T}\left(A+A^{T}\right) ∇y=xTA+xTAT=xT(A+AT) 特别地，如果A是实对称矩阵，则 ∇ y = x T A + x T A T = 2 x T A \nabla y=x^{T} A+x^{T} A^{T}=2x^{T}A ∇y=xTA+xTAT=2xTA