Eigenmatrices of a given vector. Given a vector $v$, I would like to find the set of its "eigenmatrices" - that is, the set of all matrices $A$ s.t. $Av=\lambda v$ for some constant $\lambda$.
(Following this, I would like to restrict $\lambda\to1$, but this'll do for now.)
Is there an existing name for this problem? Where might I go looking for solutions?

Let's work in $\Bbb R^n$ and have $\|v\|=1$.
One way to describe your eigenmatrices is to take $\{v\}^\bot$ - the orthogonal complement of $v$. Take $n$ vectors $w_k\in \{v\}^\bot$, $k=1..n$. Then compose a matrix $B$ such that its $k$-th row is $w_k^T$. Finally,
$$A=B+\lambda vv^T$$
is an eigenmatrix and all eigenmatrices have such a form.
The set of eigenmatrices is an affine subset of $M_n(\Bbb R)$. It's dimension is $n(n-1)$.
