Let $F$ be a field with $\operatorname{char}(F) \ne 2$, let $V$ be an $n$-dimensional [vector space](/page/Vector%20Space) over $F$, and let $q: V \to F$ be a [quadratic form](/page/Quadratic%20Form). Let $\mathcal B$ and $\mathcal C$ be ordered bases of $V$, and let $P_{\mathcal B \leftarrow \mathcal C} \in F^{n \times n}$ be the change-of-coordinates matrix satisfying

\begin{align*} [v]_{\mathcal B}=P_{\mathcal B \leftarrow \mathcal C}[v]_{\mathcal C} \end{align*}

for every $v \in V$. If $A \in F^{n \times n}$ is the matrix of $q$ in the basis $\mathcal B$, meaning that

\begin{align*} q(v)=[v]_{\mathcal B}^{\top}A[v]_{\mathcal B} \end{align*}

for every $v \in V$, then the matrix of $q$ in the basis $\mathcal C$ is

\begin{align*} P_{\mathcal B \leftarrow \mathcal C}^{\top}AP_{\mathcal B \leftarrow \mathcal C}. \end{align*}