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 constant positive real scalars and $v_i$ are **constant complex valued finite dimensional matrices** all of same dimension. So that makes $v_iv_i^H$ positive semidefinite (PSD). $I$ is the identity matrix.

Now we want to maximize the following determinant over $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ $$\mathrm{maximize}_{\{1,\dots,n\}}\:\det \left( I+\frac{a_iv_iv_i^H}{I+\sum_{j\neq i} a_jv_jv_j^H} \right).$$

Essentially we pick one matrix for the numerator and all the rest go in the denominator. Which matrix should go on the numerator?

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

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