Given eigenvectors $v_1, v_2, \dots, v_n$ and eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$, how do I construct a matrix whose eigenvectors and eigenvalues are $v$ and $\lambda$?
The straightforward way of doing this is to encapsulate all $n^2$ constraints into a linear system and solve for each element of the matrix $M_i$. I.e.,
$$ \begin{bmatrix} v_{11} & v_{12} & \dots & v_{1n} & 0 & 0 &\dots & 0 & \dots & 0 & 0 &\dots & 0 \\ 0 & 0 &\dots & 0 & v_{11} & v_{12} & \dots & v_{1n} & \dots & 0 & 0 &\dots & 0\\ & & & & & & \vdots \\ 0 & 0 &\dots & 0 & 0 & 0 &\dots & 0 & \dots & v_{n1} & v_{n2} & \dots & v_{nn}\\ \end{bmatrix} \begin{bmatrix} M_1 \\ M_2 \\ \vdots \\ M_{n^2} \end{bmatrix} = \begin{bmatrix} \lambda_1 v_{11} \\ \lambda_1 v_{12} \\ \vdots \\\lambda_n v_{nn} \end{bmatrix} $$
Is there a more elegant way?