Suppose we have a a set of matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are positive real scalars and $v_i$ are complex valued finite dimensional matrices. $I$ is the identity matrix.

Now we want to maximize the following determinant. $$\det \left( I+\frac{a_iv_iv_i^H}{I+\sum_{j\neq i} a_jv_jv_j^H} \right).$$

I think the matrix which should go on the numerator is the one with the largest eigenvalues or in the case of PD which is equal to largest determinant $\det(a_iv_iv^H_i)$. Is this claim right?

So I proceeded: $$\det \left( I+\frac{a_iv_iv_i^H}{I+\sum_{j\neq i} a_jv_jv_j^H} \right)=\det \left( I+\sum_{j} a_jv_jv_j^H \right) \det \left( \left( I+\sum_{j\neq i} a_jv_jv_j^H \right)^{-1}\right).$$ The matrices $a_iv_iv^H_i$ are PSD and have non negative eigenvalue. Could someone please help to complete the proof.

P.S.: I believe this question is related to Determinant of sum of positive definite matrices.