16
$\begingroup$

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.

$\endgroup$
22
$\begingroup$

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.

$\endgroup$
12
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ 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 '16 at 6:58
  • $\begingroup$ @abx could you write in matrix form what $J.\wedge^{n-1}A$ is? $\endgroup$ – T.... Mar 27 '18 at 13:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.