-- In quantum theory observables can be represented by Hermitian matrices.
-- If an observable of a system can be represented by a particular matrix at a particular instant, then all matrices of the same dimension represent observables of that system.
-- In a state specified by the vector |psi>, an observable X is sharp if and only if X|psi> = x|psi> for some real number x. In which case x is an eigenvalue of X and |psi> is an eigenvector of X.
Now let Y be any matrix that does not have |psi> among its eigenvectors. (For any vector, there exists an infinity of such matrices.)
If the actual state is |psi>, the observable Y cannot be sharp. (Because of the 'if and only if' above.)