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

$\Delta_x=\Delta_y$

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

Proof 显然

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

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

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

\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}