the normalized Fubini-Study metric in local coordinates then we set $$\nu = \omega^n =\, n!\, \left( \frac{\sqrt{-1}}{2\pi} \right)^n
        \frac{
            dz_1 \wedge d\overline{z}_1 \wedge \cdots \wedge dz_n \wedge d\overline{z}_n
            }{
            \left( 1 + |z_1|^2 + \ldots + |z_n|^2 \right)^{n+1}
        }.$$ It is not hard to check that with this choice of volume form we get $$\mathrm{Vol}({\mathbb{CP}}^n) = \int_{{\mathbb{CP}}^n}\!\!\! \nu = 1.$$

Again we set $L = O(1)$ and fix a $k>0$. Note that a basis of $H^0(X,L^k)$ is given by the set of monomials in the $z_i$ of total degree $\leq k$. Denote these by $w_1, \ldots, w_N$ where $N={n+k \choose k}$. In this set-up we are studying embeddings $$X = {\mathbb{CP}}^n \hookrightarrow {\mathbb{P}}\left( H^0({\mathbb{CP}}^n, O(k)) \right) \cong {\mathbb{CP}}^{N-1}.$$

As above we take $h$ to be the metric on $L^k = O(k)$ defined by $$h = \left( \sum_{i=1}^N |w_i|^2 \right)^{-1}.$$ Now if $G$ is a (positive definite Hermitian) matrix on $H^0(X,L^k)$ then ${T_{\nu}}(G)$ is the matrix giving rise to the norm $$\| s \|_{\mathrm{Hilb}_{\nu}} = R_{\nu} \int_X |s|_h^2 \nu,$$ where $$R_{\nu} = \frac{\mathrm{dim}\,  H^0(X, L^k)}{\mathrm{Vol}(X, \nu)} = N.$$

The matrix $G$ has rows and columns indexed by the terms $w_i, \ i=1, \ldots, N$. Let us take a diagonal matrix with 