  The uncertainty relation is one of the key ingredients of quantum theory.
Despite the great efforts devoted to this subject, most of the variance-based
uncertainty relations are state-dependent and suffering from the triviality
problem of zero lower bounds. Here we develop a method to get uncertainty
relations with state-independent lower bounds. The method works by exploring
the eigenvalues of a Hermitian matrix composed by Bloch vectors of incompatible
observables and is applicable for both pure and mixed states and for arbitrary
number of N- dimensional observables. The uncertainty relation for incompatible
observables can be explained by geometric relations related to the parallel
postulate and the inequalities in Horn's conjecture on Hermitian matrix sum.
Practical entanglement criteria are also presented based on the derived
uncertainty relations.
