矩阵求导
公约
若\(\bar{y}=\Psi(\bar{x})\)，其中 \(\bar{y},\bar{x}\) 分别是 \(m\times 1,n\times 1\) 向量，则有Jacobian矩阵： \[ \frac{\partial\bar{y}}{\partial\bar{x}} = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n} \\\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n} \\\\ \vdots & \vdots & \ddots & \vdots \\\\ \frac{\partial y_n}{\partial x_1} & \frac{\partial y_n}{\partial x_2} & \cdots &\frac{\partial y_n}{\partial x_n} \end{bmatrix} \]
按照该公约：如果 \(x\) 是标量，则Jacobian矩阵是 \(m \times 1\) 列向量；如果 \(y\) 是标量，则Jacobian矩阵是 $ 1 n$ 行向量。
定理 1
若 \(\bar{y}=\mathbf{A} \bar{x}\)，则: \[ \frac{\partial\bar{y}}{\partial\bar{x}}=\mathbf{A} \]
定理 2
若标量 \(\alpha\) 满足 \(\alpha =\bar{y}^T\mathbf{A} \bar{x}\)，则: \[ \frac{\partial \alpha}{\partial\bar{x}}=\bar{y}^T\mathbf{A},\; \frac{\partial \alpha}{\partial\bar{y}}=\bar{x}^T\mathbf{A}^T \]
定理 3
若标量 \(\alpha\) 满足 \(\alpha =\bar{x}^T\mathbf{A} \bar{x}\)，则: \[ \frac{\partial \alpha}{\partial\bar{x}}=\bar{x}^T \left( \mathbf{A}^T + \mathbf{A} \right) \]