Eigenvalues of the sum of a diagonal matrix and the outer product of two vectors

Question

Does a $n \times n$ matrix $M = D + u. v^T$ with $D$ diagonal, $u$ and $v$ two given vectors, and $n$ a positive integer have some interesting properties in term of spectrum (all the real parts of the eigenvalues positive, for example) ?

I'm interested in the stability of the associated system of first order linear differential equations.

If $u$ and $v$ are sufficiently small, you can get a quick-and-easy result using the Gershgorin circle theorem — Ben Grossmann, Mar 30 '15 at 14:35
We can also say nice things if $u = v$ (or they're scalar multiples of each other). — Ben Grossmann, Mar 30 '15 at 14:37
@Omnomnomnom, thanks for those answers. What about assuming that all the coefficients of $D$, $u$ and $v$ are strictly positive? — Cyanocitta, Mar 30 '15 at 14:53
I don't think that gets you far in terms of characterizing the spectrum. It does let you say something about the largest eigenvalue, though. — Ben Grossmann, Mar 30 '15 at 15:07
You also might be able to use positive definiteness in the sense described here. In particular, if $D$ has positive diagonal entries and if the symmetric $uv^T + vu^T$ is positive definite, then $D + uv^T$ will have eigenvalues with positive real part. — Ben Grossmann, Mar 30 '15 at 15:17

Algebraic Pavel · Accepted Answer · 2015-03-30T21:10:28.170

1

If $D$ is a positive definite diagonal matrix and $u:=[u_1,\ldots,u_n]^T$ and $v:=[v_1,\ldots,v_n]^T$ are positive vectors, a very useful fact is that $M:=D+uv^T$ is symmetrizable. That is, if $$\tag{1} S:=\mathrm{diag}\left(\sqrt{\frac{u_i}{v_i}}\right)_{i=1}^n, $$ then $$ S^{-1}MS=D+ww^T, \quad w:=S^{-1}u=Sv, $$ is symmetric. Hence all eigenvalues of $M$ are real (and the associated eigenvectors are real as well).

Now since $ww^T$ is semidefinite and $D$ is positive definite, $D+ww^T$ is positive definite and the eigenvalues of $D+ww^T$ (and hence the eigenvalues of $M$) are positive.

Note that the assumption on the positivity of $u$ and $v$ can be relaxed. For each $i=1,\ldots,n$, we might require either that $u_i$ and $v_i$ have the same sign (in this case, (1) is well-defined) or they are both zero (in this case, the corresponding diagonal entry of $S$ can be arbitrary nonzero).

edited Mar 30 '15 at 21:10

answered Mar 30 '15 at 15:55

Algebraic Pavel

22,928

That's a great answer, thanks! – Cyanocitta Mar 30 '15 at 20:20
Is there a closed-form solution to give the eigenvalues of $D+w.w^T$ ? – user3433489 Jan 28 '21 at 03:34
@user3433489 Not really. – Algebraic Pavel Jan 28 '21 at 12:28

Eigenvalues of the sum of a diagonal matrix and the outer product of two vectors

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