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eigenvalues of matrices or operators

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Eigenvalues of a partitioned self-adjoint matrix

Assume a positive-definite $2\times2$ matrix ($a_3=a_2^*$) $$ A=\pmatrix{a_1 & a_2\\a_3 & a_4} $$ whose eigenvalues are known. … In particular: Is there an algorithm for calculating the eigenvalues of $A'$ from the eigenvalues of $A$ and $B_i$? …
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