usually eliminated by adding appropriate gauge-fixing terms. If we require the $R_\xi$ gauge-fixing procedure remove all the vector-scalar two-point transitions, then it is natural to consider adding the following gauge-fixing Lagrangian: $$\mL_{gf}=-\sum_{p=1}^{n_M}\frac{1}{2\xi^p}(\partial_\mu Z_p^\mu-\xi^p\tilde{G}_p)^2
\label{eq:gf}$$ Here $\xi^p,p=1,2,...,n_M$ are gauge parameters. There is freedom in the choice of the gauge-fixing function and the requirement to remove vector-scalar two-point transitions is not sufficient to uniquely determine it. However we will see below there is a theoretically well-motivated choice which facilitates the diagonalization process. After adding the gauge-fixing terms, we have $$\begin{aligned}
& \mL_{quad}+\mL_{gf}\supset \nonumber \\
& \frac{1}{2}V_{ij}(\partial_\mu G_i)(\partial^\mu G_j)
-\frac{1}{2}\xi^p\tilde{G}_p^2-\frac{1}{2}(\bM_G^2)_{ij}G_i G_j
-\frac{1}{2\xi^p}(\partial_\mu Z_p^\mu)^2+\frac{1}{2}(\bM_V^2)_{pq}Z_{p\mu}Z_q^\mu\end{aligned}$$

The matrix $V$ denotes the scalar kinetic matrix. If it is not the identity matrix, we may simply use the complete-the-square method to diagonalize it and then make the resulting terms canonically normalized. This is in complete analogy to the diagonalization of quadratic forms in linear algebra. Note that the overall transformation employed to render the scalar kinetic terms canonically normalized need not be orthogonal.

Now suppose we have found a transformation of the scalar fields $$S_i=U_{ij}G_j
\label{eq:sug}$$ which renders the scalar kinetic terms diagonalized and canonically normalized: $$\frac{1}{2}V_{ij}(\partial_\mu G_i)(\partial^\mu G_j)
=\frac{1}{2}(\partial_\mu S_i)(\partial^\mu S_i)$$ Here $U$ is a real invertible 