Consider a column vector $v\in \mathbb{R}^n$. We are interested in finding the pseudo-inverse of the following matrix: \begin{align} A= (I_n \otimes v^T) + (v^T \otimes I_n) \end{align} where $I_n$ is the identity matrix of dimension $n$ and $\otimes$ is the Kronecker product.
Recall, that the pseudo-inverse (left inverse) of a matrix $A$ is defined as \begin{align} A^{+}=A^T (AA^T)^{-1} \end{align}
What I tried Clearly, if $n=1$ we have that $A^{+}$ is equal to $\frac{1}{2v}$.
I also computed $A^TA$ and tried to find the inverse of it but with no luck.