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Say that you have an eigenvalue problem of a Hermitian matrix ##A## and want (for many reasons) to calculate the eigenvalues and eigenstates for many cases where only the diagonal elements are changed in each case.

Say the common eigenvalue problem is ##Ax=λx##. The ##A## matrix is a sum of two matrices ##A=B+C## and the ##C## matrix contains only the diagonal elements non-zero. For example

\begin{equation}

\begin{pmatrix}

E_{1}& a & b \\

c & E_{2}& d \\

e & f & E_{3}

\end{pmatrix}

=

\begin{pmatrix}

0& a & b \\

c& 0 & d \\

e& f & 0

\end{pmatrix}

+

\begin{pmatrix}

E_{1}& 0 & 0 \\

0& E_{2} & 0 \\

0& 0 & E_{3}

\end{pmatrix}

\end{equation}

Say you want to calculate the eigenstates of the ##A## matrix but in a different way.

I was thinking is there is a method which is used to split the problem in two other sub-problems. The first step to be something like to calculate something for the ##B## matrix which remains unchanged in each case so the calculation is performed only one time, in the second step you only perform calculations for the ##C## matrix which in any case onlyn the diagonal elements are changed so you have to work only with the ##C## matrix problem and finally you combine the data of these two matrices in order to obtain the final result for the ##A## matrix.

Whats your opinion?

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# A Eigenstates of "summed" matrix

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