For $\mathbf{x}\in\mathbb{R}^n$ consider the linear system $$\mathbf{b}=A\mathbf{x},$$ where $A$ is square. I would like to study the spectral properties of $A+\sum \alpha_i A_i$ for all $A_i$ such that $A_i\mathbf{x}=0$ (I.e., $\mathbf{x}\in\mathrm{null}(A_i)$). In other words, how do the eigenvalues and eigenvectors of $A$ change when added to linear combinations of ${A_i}$ such that the resulting matrix leaves the solution of the linear system invariant?
Is there a special name for these matrices or for this problem?
E.g., for $n=3$ we have $\mathbf{x} = (x,y,z)^T\in\mathbb{R}^3$ and the matrices $A_i$ are from the following set: $$ A_i \in \left\{ \left[ \begin {array}{ccc} 0&0&0\\ 0&0&0\\ -y&x&0\end {array} \right] , \left[ \begin {array}{ccc} 0&0&0\\ 0&0&0\\ -z&0&x\end {array} \right] , \left[ \begin {array}{ccc} 0&0&0\\ 0&0&0\\ 0&-z&y\end {array} \right] , \left[ \begin {array}{ccc} -y&x&0\\ 0&0&0\\ 0&0&0\end {array} \right] , \left[ \begin {array}{ccc} -z&0&x\\ 0&0&0\\ 0&0&0\end {array} \right] , \left[ \begin {array}{ccc} 0&-z&y\\ 0&0&0\\ 0&0&0\end {array} \right] , \left[ \begin {array}{ccc} 0&0&0\\ -y&x&0\\ 0&0&0\end {array}\right] , \left[ \begin {array}{ccc} 0&0&0\\ -z&0&x\\ 0&0&0\end {array} \right] , \left[ \begin {array}{ccc} 0&0&0\\ 0&-z&y\\ 0&0&0\end {array} \right] \right\} $$