Eigenmatrices of a given vector.

Question

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?

Answer 1

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 $n(n-1)$.

Answer 2

Hint: $\lambda$ doesn't make much difference. Try writing the equation $Av=\lambda v$ in terms of the entries of $A$. Let us say $\mathbf{x}$ is the vector with all the columns of $A$ stacked up in a single large column. Try to see now, if you can come up with a set of linear equations in the form $B\mathbf{x}=\lambda v$ where $B$ is a matrix with zeros with certain positions and entries of $v$ in others. Now think about when this linear system will have a solution.