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Hi I have a question towards how to expand the gaps between eigenvalues for one specific matrix.

  1. Suppose that we have a matrix ${\bf A}$, and the eigenvalues are $\{\lambda_1, \lambda_2, ..., \lambda_k\}$, suppose that the gaps between each consecutive eigenvalues are denoted by $\{\Delta_{1,2}, \Delta_{2,3},...\Delta_{k-1,k}\}$, and now I want to make some transformation on $\bf A$. Suppose we use matrix perturbation, and $\bf{B} = \bf{A} + \Delta$, and the minimum gaps for $\bf B$ is larger than some specific value say $\alpha$. So Does anyone have some ideal how to do this?

  2. If I already have the transformation method, and I have matrix $\bf B$, I calculate all eigenvalues(eigen-pairs) of $\bf B$, can I recover matrix $\bf A$'s eigenvalues(eigen-pairs)?

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    Are $A$ and $B$ symmetric? (This sort of thing behaves somewhat badly in the nonsymmetric case.) Also, should $\Delta$ be small? How small if so? – 2017-01-25
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    I don't see any way to reliably do this without having some information about the eigenvalues/eigenvectors of $A$ a-priori – 2017-01-25
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    Hi, A and B are symmetric, and I do not have restrictions on $\bf \Delta$, thanks – 2017-01-25

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