First let me make two statements to give my question the proper context.
Consider $D$ a diagonal matrix and $u v^T$ a rank 1 matrix. From my current knowledge there exist computationally cheap algorithms to diagonalize $D+uv^T$ as discussed here: maximum eigenvalue of a diagonal plus rank-one matrix
Diagonalizing a matrix of the form $Duv^T$ is straightforward, see here: Eigenvector of a diagonal matrix times a rank-1 matrix
Now to my question(s):
I want to diagonalize $D^2+Duv^T$. Is $Duv^T$ still a rank 1 matrix? I considered some examples but I am not really sure if this is a general statement.
This question depends on the answer of the first. If the statement is true the algorithms in statement 1 should be applicable to diagonalization of $D^2+Duv^T$, no? If it is not true are there others ways to tackle this?