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Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more generally $A_t := A + tJ$.

Is there a nice description of the eigenvalues or eigenvectors of $A_t$ in terms of those of $A$? If not, what about for $t$ small?

It would be great to have an answer for general coefficient fields, but I would also be interested in the case with $A \in M_n(\mathbb{C})$ or $A \in M_n(\mathbb{R})$ with all nonnegative or positive entries, if they have nicer answers.

Thank you.

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This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.

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A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(x).$$ The proof only needs the fact that a determinant doesn't change when a multiple of one row is added to a different row, so I don't see why it wouldn't be true for all fields.

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    $\begingroup$ In other words, $\det(A+sJ)=\det(A)+s\mathrm{Tr}(J\cdot \wedge^{n-1}A)$ is an affine function of $s$. $\endgroup$
    – abx
    Jul 7, 2016 at 6:58
  • $\begingroup$ @abx could you write in matrix form what $J.\wedge^{n-1}A$ is? $\endgroup$
    – Turbo
    Mar 27, 2018 at 13:04

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