Let $A,\,B\in\mathbb{R}^{n\times n}$ be tridiagonal matrices with constant elements with eigenvalues $\lambda_i$ and $\mu_i$, for $i=1,\ldots,n,$ respectively. Let the matrix $C=B^{-1}A.$ The matrices $A$ and $B$ doesn't have the same eigenvectors, so in general the eigenvalues of $C$ is not the $\frac{\lambda_i}{\mu_i}.$ Is any way to estimate the eigenvectors of matrix $C$ if we have a closed formula for eigenvalues and eigenvector of matrices $A$ and $B$?

  • $\begingroup$ You must know that the inverse of a tridiagonal matrix can be very dense. Thus, I the hope to have an estimation of this kind is very small. $\endgroup$ – Jean Marie Nov 27 '18 at 19:04

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