随笔 - Laplace 算子的旋转不变性

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一道练习题

定理 (Laplace 算子的旋转不变性)

令

  • \(O=(o_{ij})_{n\times n}\in\mathbb{R}^{n\times n}\) 是正交矩阵
  • \(x=(x_1,x_2,\dots,x_n)^T\in\mathbb{R}^n\)
  • \(y=Ox=(y_1,y_2,\dots,y_n)^T\in\mathbb{R}^n\)

则

\[ \Delta_x=\Delta_y \]

其中

  • \[ \Delta_x:=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2} \]
  • \[ \Delta_y:=\sum_{i=1}^n\frac{\partial^2}{\partial y_i^2} \]

Proof 显然

\[ y_i=\sum_{j=1}^no_{ij}x_j,\ \forall i=1..n \]

故我们有

\[ \frac{\partial}{\partial x_i}=\sum_{j=1}^n\frac{\partial}{\partial y_j}\frac{\partial y_j}{\partial x_i}=\sum_{j=1}^no_{ji}\frac{\partial}{\partial y_j} \]

考虑 Hessian 矩阵

\[ \nabla^2_x=\left(\frac{\partial^2}{\partial x_j\partial x_i}\right)_{n\times n}=\left(\sum_{l=1}^n\sum_{k=1}^no_{lj}o_{ki}\frac{\partial^2}{\partial y_l\partial y_k}\right)_{n\times n} \]

则

\[ \begin{aligned} \Delta_x&=\operatorname{tr}(\nabla^2_x)\\ &=\sum_{l=1}^n\sum_{k=1}^n\left(\sum_{i=1}^no_{li}o_{ki}\right)\frac{\partial^2}{\partial y_l\partial y_k}\\ &=\sum_{l=1}^n\sum_{k=1}^n\delta_{kl}\frac{\partial^2}{\partial y_l\partial y_k}\\ &=\sum_{k=1}^n\frac{\partial^2}{\partial y_k^2}\\ &=\Delta_y \end{aligned} \]