矩阵微分

矩阵 微分-导数-偏导-梯度 index

---

乘法

$$(Ax)_i = \Sigma_k a_{ik}x_k.$$

$$\quad (y^\intercal A)_i = \Sigma_k a_{ki}y_k.$$

$$(y^\intercal Ax) = \Sigma_j y_j(\Sigma_k a_{jk}x_k) = \sum_{j,k} a_{jk}y_j x_k.$$

微分

$$\frac{\partial(Ax)}{\partial x} = A.$$

$$\frac{\partial(y^\intercal Ax)}{\partial x} = y^\intercal A, \quad \frac{\partial(y^\intercal Ax)}{\partial y} = (Ax)^\intercal.$$

$$\frac{\partial(x^\intercal Ax)}{\partial x} = x^\intercal A + (Ax)^\intercal \stackrel{A^\intercal = A}{=} 2x^\intercal A.$$