Eigenvectors of a linear combination of symmetrical matrices

Assume we know the eigenvectors of a matrix A. Let us take a matrix N:

N=aA-bE

(a,b a real values, E is identity matrix, A is a tridiagonal matrix).

I veryfied numerically that the eigenvectors of N are the same like eigv. of A.

How to explain this? I suspect that such a case is well known in algebra field, a need to write a neat statement, like: "it is well known from linear algebra that eig(N)=eig(A), becouse ..... "

Tom

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    ... Tomek wrote: ... (a,b a real values, E is identity matrix, A is a tridiagonal matrix). ... I veryfied numerically that the eigenvectors of N are the same like eigv. ... I suspect that such a case is well known in algebra ...
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