# Approximating the eigenvalues and eigenvectors of $B := A + E$

Suppose I have a matrix $$B := A + E$$, where $$A$$ is diagonal and $$E$$ is an off-diagonal, symmetric matrix whose non-diagonal elements are small. Is there any way to obtain the approximate eigenvalue and eigenvector of $$B$$?

I came across this approximation

$$\lambda_B = \lambda_A + \frac{x^t E x}{x^t x}$$

where $$x$$ is the eigenvector of $$A$$. However, this does not work for me, because, $$A$$ is diagonal, and diagonal elements of $$E = 0$$. In this case, I get $$x^t E x = 0$$. Is there any other method?

Also assume the eigenvalues of $$\lambda_A$$ is positive, if it helps.

• Eigenvalue? Singular? May 20, 2022 at 12:53
• Off-diagonal matrix means here a matrix with zero-diagonal.
– KBS
May 20, 2022 at 13:13
• Edited the question in which I added that E is symmetric. Assume the diagonal entries of A is positive, if it helps. May 20, 2022 at 15:38
• The eigenvalues should not depend on $x$. May 20, 2022 at 15:50
• I would rewrite $A:= D+E$ where $D$ is diagonal. May 20, 2022 at 15:55