Construct matrix given eigenvalues and eigenvectors

Solution 1:

Your system of equations is $Mv_1=\lambda_1v_1,\ldots,Mv_n=\lambda_nv_n$. Or equivalently, $M(v_1,\ldots,v_n)=(\lambda_1 v_1,\ldots,\lambda_n v_n)$, where $V:=(v_1,\ldots,v_n)$ is the $n\times n$-matrix with columns $v_1,\ldots,v_n$. You can write this as $MV=VD$ where $D$ is the diagonal matrix with diagonal entries $\lambda_1,\ldots,\lambda_n$. So, assuming $V$ is invertible, that is, that your given eigenvectors are linearly independent, you get $M=VDV^{-1}$. Thus to calculate $M$ this way, all you need to do is to find the inverse of the matrix of eigenvectors, and multiply three matrices together.