While reading the definition of pseudo determinant in here, I've found the following :
$$pdet(L) : = \sum_{I\in[n] , |I| = r}det(L_{I,\ I}) = \prod_{i=1}^r\lambda_i$$
where $L_{I,\ I}$ denotes the principal minor, which strikes out two same indexed rows and columns from the original matrix.
The author presented the second equation very trivially, but I can't prove the relationship between the sum of all principal minors and the mulitplication of eigenvalues.