Say you're given a set of eigenvalues and eigenvectors, is it always possible to determine the original matrix A?
I recognize that if a matrix A is diagonalizable, the geometric multiplicity equals the algebraic multiplicity and you can form you matrix of Eigenvectors, call it Q, and then your diagonal matrix D (which consists of the eigenvalues that correspond to the correct eigenvector column in Q) gives you the result
$A = QDQ^-1$
but assuming we aren't given the original matrix A, is it possible to "reverse" this process and find A? Thanks