Reduction of the rank of a symmetric real matrix by adding a pure antisymmetric imaginary matrix?

Hi,

A is a given NxN real symmetric matrix, which has eigenvalues l_i (|
l_1|>= |l_2|>=...>= |l_N|). B is a pure imaginary antisymmetric NxN
matrix.

Let C=A+B, which is Hermitian. We denote C's eigenvalues as u_i (|u_1|
=|u_2|>=...>=|u_N|).

Given L, we are looking for B such that \sum_{i=L}^N |u_i|^2 reach the
minimum.

I'm wondering if there is any efficient way of solving this problem?

Thanks,
Peng

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