Eigenstates of "summed" matrix

[ilvreth]

Hi to all.

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?

 

[andrewkirk]

If the diagonal elements are only changed by adding the same amount to all three then there is a simple solution. First solve the eigenvalue problem for $A$. Then the eigenvalues for the matrix that is $A$ with $h$ added to all diagonal elements are simply the eigenvalues of $A$ with $h$ added to them. The eigenvectors are unchanged by the addition of $h$ along the diagonal.

For more general changes to the diagonal elements, I see no reason to expect there to be a solution that is any easier than just re-solving the eigenvalue problem in each case.