Eigenvalues of the sum of a diagonal matrix and the outer product of two vectors 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 $D$ 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).
