设\(\alpha_1,\alpha_2,\cdots ,\alpha_n\)、\(\beta_1,\beta_2,\cdots ,\beta_n\)是\(V\)的两组基，且
\begin{equation*}
(\alpha,\alpha_2,\cdots ,\alpha_n)=(\beta_1,\beta_2,\cdots ,\beta_n)P.
\end{equation*}
设\(A\)是基\(\alpha_1,\alpha_2,\cdots ,\alpha_n\)的度量矩阵，\(B\)是基\(\beta_1,\beta_2,\cdots ,\beta_n\)的度量矩阵，即
\begin{equation*}
A=\left((\alpha_i,\alpha_j)\right)_{n\times n},\ B=\left((\beta_i,\beta_j)\right)_{n\times n},
\end{equation*}
下证\(A\)合同于\(B\)。 设\(\xi_1,\xi_2,\cdots ,\xi_n\)是\(V\)的一个标准正交基，且
\begin{equation*}
(\alpha_1,\alpha_2,\cdots \alpha_n)=(\xi_1,\xi_2,\cdots ,\xi_n)Q,\ (\beta_1,\beta_2,\cdots ,\beta_n)= (\xi_1,\xi_2,\cdots ,\xi_n)R,
\end{equation*}
则\(Q=RP\)。记
\begin{equation*}
R=(\gamma_1,\gamma_2,\cdots ,\gamma_n),\ Q=(\kappa_1,\kappa_2,\cdots ,\kappa_n),
\end{equation*}
其中\(\gamma_i,\kappa_i\in\mathbb{R}^n(i=1,2,\cdots ,n)\)，则
\begin{equation*}
(\alpha_i,\alpha_j)=\kappa_i^T \kappa_j,\ (\beta_i,\beta_j)=\gamma_i^T \gamma_j\mbox{。}
\end{equation*}
于是，
\begin{equation*}
A=\begin{pmatrix}
\kappa_1^T \kappa_1&\kappa_1^T \kappa_2&\cdots&\kappa_1^T \kappa_n\\
\kappa_2^T \kappa_1&\kappa_2^T \kappa_2&\cdots&\kappa_2^T \kappa_n\\
\cdots&\cdots&\cdots&\cdots\\
\kappa_n^T \kappa_1&\kappa_n^T \kappa_2&\cdots&\kappa_n^T \kappa_n
\end{pmatrix}=\begin{pmatrix}
\kappa_1^T\\\kappa_2^T\\\vdots\\\kappa_n^T
\end{pmatrix}\begin{pmatrix}
\kappa_1^T&\kappa_2^T&\cdots&\kappa_n^T
\end{pmatrix}=Q^TQ,
\end{equation*}
同理，\(B=R^TR\)。由\(Q=RP\)可知，
\begin{equation*}
A=(RP)^T(RP)=P^T(R^TR)P=P^TBP\mbox{。}
\end{equation*}
因此\(A\)合同于\(B\)。
