Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables and $|\boldsymbol{z}|^2=\sum_{i=1}^n |z^i|^2.$

My GOAL is to show that, for every $k=1,\ldots,n,$ $$ \det \left( g_{i\overline{j}} \right)_{1 \leq i,\overline{j}\leq k} = \frac{1+\sum_{i=k+1}^n |z^i|^2}{(1+|\boldsymbol{z}|^2)^{k+1}}.$$

Let's write numerator matrix as $B_{i\bar{j}}=\delta_{ij}(1+|z|^2)-z_i\bar z_j$, where $|z|^2=\sum\limits_{i=1}^k|z_i|^2$. Notice that the first component $B^1_{i\bar j}={\delta_{ij}(1+|z|^2)}$ is a scalar multiply an identity matrix, and the second component $B^2_{i\bar j}=z_i\bar z_j$ is a symmetric matrix of rank one, and $B^1B^2=B^2B^1$, so there exists invertile matrix $P$, such that $PBP^{-1}= \left( \begin{array}{ccc} 1+|z|^2 \\ & \ddots & \\ & & 1+|z|^2 \end{array} \right)- \left( \begin{array}{cccc} |z|^2 & &0 &\\ & \ddots & & \\ 0& &0 \end{array} \right)= \left( \begin{array}{ccc} 1 \\ &1+|z|^2 & \\ &\ddots& \\ & & 1+|z|^2 \end{array} \right)$ so $\det(g_{i\bar j})=\frac{1}{(1+|z|^2)^{k+1}}$.