Assume a positive-definite $2\times2$ matrix $$ A=\pmatrix{a_1 & a_2\\a_3 & a_4} $$ whose eigenvalues are known ($a_3=a_2^*$). Now let's have four different complex $d\times d$ matrices $B_1,B_2,B_3,B_4$ (self-adjoint but not necessarily positive-definite), whose eigenvalues are also known. Can there be said something about the eigenvalues of the following $2d\times 2d$ matrix $$ A'=\pmatrix{a_1B_1 & a_2B_2\\a_3B_3 & a_4B_4} $$ ?

In particular:

Is there an algorithm for calculating the eigenvalues of $A'$ from the eigenvalues of $A$ and $B_i$?

Can it be simplified if $B_2=B_3$, so $A'$ is self-adjoint?


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