The determinant of the new, reduced alpha block is simply it's (i, j ′ )-minor, the determinant of the original block after eliminating its i row and j ′ column. Up to factors of (-1), one recalls that all minors are contained in the cofactor matrix of the original alpha block. Defining the original alpha block as A ij ′ = ⟨i|j ′ ⟩, then the cofactor matrix is (A -1 ) T • det(A)/(-1) i+j ′ . The factors of (-1) are canceled by virtue of Eq. ( ) and we denote as G, the cofactor matrix modulo sign: Notice that if we rewrite the new beta block for each overlap, Φ ā i Φ b′ j ′ , V is simply a scalar and D and C are a single row and column respectively. B is the original beta block and is a constant for all excitations. Using the identity for determinants of square matricies from