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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.

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

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