# Estimate eigenvectors and eigenvalues of product of two matrices

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 $$B$$?

• 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. – Jean Marie Nov 27 '18 at 19:04