I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in  real eigenvalues and eigenvectors due to the relations between the off-diagonal coupling matrices. The eigenvalue problem reads as,$$A \phi = \lambda B \phi$$where$$ A = \begin{bmatrix} K_1 &K_2\\ 0 &K_3\end{bmatrix}$$and$$ B = \begin{bmatrix} M_1 &0\\ -K_2^T &M_2\end{bmatrix}$$I am using the Arnoldi technique to solve this eigenvalue problem. However, I always have to apply a scaling in order to find the right eigenvalues and eigenvectors. What I am applying is the simple jacobi scaling such asP1 = diag(1./sqrt(diag(K1)))P2 = diag(1./sqrt(diag(K2)))P = diag(P1,P2)eventually, what I am solving is the generalized eigenvalue problem with this scaling matrix, namely,$$ P A P = \begin{bmatrix}P_1 K_1 P_1 &P_1 K_2 P_2\\0 &P_2 K_3 P_2\end{bmatrix}$$ and $$ P B P = \begin{bmatrix}P_1 M_1 P_1 &0\\-P_2 K_2^T P_1 &P_2 M_2 P_2\end{bmatrix}$$then I go back to the original space before scaling. It is fine up to this point but there is one more follow up step I would like to accomplish. Later on, I need to solve a linear system which involves $A$, such as,$$A x = b$$and I would like to use the factor I generated during the above eigenvalue solution in order to perform this solution. But if I perform the scaling, my original $A$ transforms to $PAP$ so I can not use that for a forward-backward solve and I need a new factorization for $A$. And as mentioned above, I can only find the right eigenvalues and eigenvectors with this scaling although my original matrices are not that ill-conditioned. I am looking for a way to solve my eigenvalue problem without scaling and use the available factorization at a later stage, but so far, I have no luck without scaling. I am using the built in solvers (or the Factorize package from Tim Davies) in my MATLAB implementations for the Arnoldi iterations.Any help/idea is greatly appreciated on what I can try to solve this problem without scaling.
Scaling for a nonsymmetric eigenvalue problem
linear solver;eigensystem;preconditioning