need to deal with. It is necessary to ensure that the matrix $K_M$ inferred from Eq.  is indeed an orthogonal matrix, otherwise we will not be able to keep the scalar kinetic terms in their diagonalized and canonically normalized form.

To help determine whether the matrix $K_M$ inferred from Eq.  is orthogonal we denote the real vector space spanned by $G_i,i=1,2,...,n_S$ as $\mathbb{L}$ and introduce an inner product in $\mathbb{L}$, defined by $$\langle S_i|S_j\rangle\equiv\delta_{ij},i,j=1,2,...,n_S$$ This means the $S_i$’s constitute an orthonormal basis in $\mathbb{L}$. The inner product of any two elements in $\mathbb{L}$ can then be calculated by virtue of the linearity property of the inner product. It is obvious that the $\bar{S}_i$’s also form an orthonormal basis in $\mathbb{L}$. Based on simple algebraic knowledge the problem of judging whether $K_M$ is orthogonal reduces to judging whether $\tilde{S}_p,p=1,2,...,n_M$ form an orthonormal basis in the subspace spanned by themselves.

As long as all the $\tilde{G}_p$’s have positive norm, we may always adjust the $\alpha_p$’s so that $$\langle\tilde{S}_p|\tilde{S}_p\rangle=1,\quad\forall p=1,2,...,n_M$$ Therefore the question becomes whether $\langle\tilde{S}_p|\tilde{S}_q\rangle=0$ holds when $p,q=1,2,...,n_M$ and $p\neq q$. According to Eq.  we only need to check whether $\langle\tilde{G}_p|\tilde{G}_q\rangle=0$ holds when $p,q=1,2,...,n_M$ and $p\neq q$.

Fortunately, when the scalar fields 