This is taken from the article here:
https://www.tandfonline.com/doi/abs/10.1080/03081087.1982.11882087
Given a matrix $$C = \mathrm{diag}(\alpha_1,\alpha_2,...,\alpha_n) + V\,\mathrm{diag}(\beta_1,\beta_2,...,\beta_n)V^*$$
where $V$ is an $n\times n$ unitary matrix.
The author uses the Cauchy-Binet formula to obtain \begin{gather} \det C = \\ \sum_{t=0}^{n} \sum_{1\le j_1 \lt...\lt j_{n-t} \le n} \sum_{1\le k_1 \lt...\lt k_{t} \le n} \Big|\det V[i_1,i_2,...,i_t|k_1,k_2,...k_t]\Big|^2\alpha_{j_1}\alpha_{j_2}...\alpha_{j_{n-t}}\beta_{k_1}\beta_{k_2}...\beta_{k_t} \end{gather}
where $\{j_1,...j_{n-t}\}$ and $\{i_1,...i_t\}$ are complementary in $\{1,2,...,n\}$
I don't see how the author is getting this result. The Cauchy Binet formula concerns the product of two matrices... so which two matrices are being used to apply the Cauchy Binet formula? Can someone break this down for me in steps? Thanks.